# Complete Odd-Prime Three-Valued Classification and Method Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261112-r06-legendre-symbol-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=3 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(3, [[0, 0], [1, 1], [2, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (3, [[0, 0], [1, 1], [2, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0001/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 2. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=5 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(5, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (5, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0002/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 3. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=7 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(7, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (7, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0003/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 4. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=11 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(11, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (11, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0004/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 5. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=13 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(13, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (13, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0005/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 6. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=17 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(17, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (17, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0006/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 7. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=19 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(19, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (19, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0007/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 8. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=23 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(23, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (23, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0008/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 9. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=29 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(29, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, 1], [21, -1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (29, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, 1], [21, -1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0009/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 10. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=31 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(31, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, 1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, 1], [29, -1], [30, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (31, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, 1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, 1], [29, -1], [30, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0010/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 11. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=37 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(37, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, -1], [30, 1], [31, -1], [32, -1], [33, 1], [34, 1], [35, -1], [36, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (37, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, -1], [30, 1], [31, -1], [32, -1], [33, 1], [34, 1], [35, -1], [36, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0011/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 12. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=41 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(41, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, 1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (41, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, 1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0012/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 13. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=43 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(43, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (43, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0013/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 14. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=47 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(47, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, 1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, 1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, 1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, 1], [43, -1], [44, -1], [45, -1], [46, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (47, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, 1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, 1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, 1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, 1], [43, -1], [44, -1], [45, -1], [46, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0014/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 15. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=53 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(53, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, 1], [44, 1], [45, -1], [46, 1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (53, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, 1], [44, 1], [45, -1], [46, 1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0015/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 16. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=59 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(59, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, 1], [46, 1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, 1], [54, -1], [55, -1], [56, -1], [57, 1], [58, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (59, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, 1], [46, 1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, 1], [54, -1], [55, -1], [56, -1], [57, 1], [58, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0016/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 17. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=61 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(61, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, -1], [18, -1], [19, 1], [20, 1], [21, -1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, -1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, -1], [41, 1], [42, 1], [43, -1], [44, -1], [45, 1], [46, 1], [47, 1], [48, 1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, -1], [55, -1], [56, 1], [57, 1], [58, 1], [59, -1], [60, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (61, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, -1], [18, -1], [19, 1], [20, 1], [21, -1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, -1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, -1], [41, 1], [42, 1], [43, -1], [44, -1], [45, 1], [46, 1], [47, 1], [48, 1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, -1], [55, -1], [56, 1], [57, 1], [58, 1], [59, -1], [60, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0017/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 18. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=67 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(67, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, -1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 1], [27, -1], [28, -1], [29, 1], [30, -1], [31, -1], [32, -1], [33, 1], [34, -1], [35, 1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1], [41, -1], [42, -1], [43, -1], [44, -1], [45, -1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, 1], [57, -1], [58, -1], [59, 1], [60, 1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (67, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, -1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 1], [27, -1], [28, -1], [29, 1], [30, -1], [31, -1], [32, -1], [33, 1], [34, -1], [35, 1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1], [41, -1], [42, -1], [43, -1], [44, -1], [45, -1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, 1], [57, -1], [58, -1], [59, 1], [60, 1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0018/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 19. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=71 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(71, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, 1], [30, 1], [31, -1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, -1], [56, -1], [57, 1], [58, 1], [59, -1], [60, 1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, -1], [68, -1], [69, -1], [70, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (71, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, 1], [30, 1], [31, -1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, -1], [56, -1], [57, 1], [58, 1], [59, -1], [60, 1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, -1], [68, -1], [69, -1], [70, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0019/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 20. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=73 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(73, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, -1], [21, -1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, -1], [35, 1], [36, 1], [37, 1], [38, 1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, -1], [46, 1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, 1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, 1], [70, 1], [71, 1], [72, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (73, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, -1], [21, -1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, -1], [35, 1], [36, 1], [37, 1], [38, 1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, -1], [46, 1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, 1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, 1], [70, 1], [71, 1], [72, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0020/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 21. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=79 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(79, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, -1], [25, 1], [26, 1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, 1], [47, -1], [48, -1], [49, 1], [50, 1], [51, 1], [52, 1], [53, -1], [54, -1], [55, 1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, -1], [70, -1], [71, -1], [72, 1], [73, 1], [74, -1], [75, -1], [76, 1], [77, -1], [78, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (79, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, -1], [25, 1], [26, 1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, 1], [47, -1], [48, -1], [49, 1], [50, 1], [51, 1], [52, 1], [53, -1], [54, -1], [55, 1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, -1], [70, -1], [71, -1], [72, 1], [73, 1], [74, -1], [75, -1], [76, 1], [77, -1], [78, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0021/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 22. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=83 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(83, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1], [43, -1], [44, 1], [45, -1], [46, -1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, -1], [54, -1], [55, -1], [56, -1], [57, -1], [58, -1], [59, 1], [60, -1], [61, 1], [62, -1], [63, 1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, 1], [70, 1], [71, -1], [72, -1], [73, -1], [74, -1], [75, 1], [76, -1], [77, 1], [78, 1], [79, -1], [80, -1], [81, 1], [82, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (83, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1], [43, -1], [44, 1], [45, -1], [46, -1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, -1], [54, -1], [55, -1], [56, -1], [57, -1], [58, -1], [59, 1], [60, -1], [61, 1], [62, -1], [63, 1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, 1], [70, 1], [71, -1], [72, -1], [73, -1], [74, -1], [75, 1], [76, -1], [77, 1], [78, 1], [79, -1], [80, -1], [81, 1], [82, -1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0022/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 23. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=89 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(89, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, -1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, 1], [68, 1], [69, 1], [70, -1], [71, 1], [72, 1], [73, 1], [74, -1], [75, -1], [76, -1], [77, -1], [78, 1], [79, 1], [80, 1], [81, 1], [82, -1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (89, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, -1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, 1], [68, 1], [69, 1], [70, -1], [71, 1], [72, 1], [73, 1], [74, -1], [75, -1], [76, -1], [77, -1], [78, 1], [79, 1], [80, 1], [81, 1], [82, -1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0023/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 24. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=97 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(97, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, 1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, 1], [45, -1], [46, -1], [47, 1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, 1], [55, -1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, 1], [62, 1], [63, -1], [64, 1], [65, 1], [66, 1], [67, -1], [68, -1], [69, -1], [70, 1], [71, -1], [72, 1], [73, 1], [74, -1], [75, 1], [76, -1], [77, -1], [78, -1], [79, 1], [80, -1], [81, 1], [82, -1], [83, -1], [84, -1], [85, 1], [86, 1], [87, -1], [88, 1], [89, 1], [90, -1], [91, 1], [92, -1], [93, 1], [94, 1], [95, 1], [96, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (97, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, 1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, 1], [45, -1], [46, -1], [47, 1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, 1], [55, -1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, 1], [62, 1], [63, -1], [64, 1], [65, 1], [66, 1], [67, -1], [68, -1], [69, -1], [70, 1], [71, -1], [72, 1], [73, 1], [74, -1], [75, 1], [76, -1], [77, -1], [78, -1], [79, 1], [80, -1], [81, 1], [82, -1], [83, -1], [84, -1], [85, 1], [86, 1], [87, -1], [88, 1], [89, 1], [90, -1], [91, 1], [92, -1], [93, 1], [94, 1], [95, 1], [96, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0024/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 25. Legendre Classes Include Zero before Nonzero Residue Classification

Use bounded odd prime p=101 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

View answer and explanation(101, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, 1], [15, -1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, 1], [55, -1], [56, 1], [57, -1], [58, 1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, -1], [70, 1], [71, 1], [72, -1], [73, -1], [74, -1], [75, -1], [76, 1], [77, 1], [78, 1], [79, 1], [80, 1], [81, 1], [82, 1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1], [89, -1], [90, -1], [91, -1], [92, 1], [93, -1], [94, -1], [95, 1], [96, 1], [97, 1], [98, -1], [99, -1], [100, 1]], Yes, zero_residue_nonresidue_partition_is_complete)

Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (101, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, 1], [15, -1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, 1], [55, -1], [56, 1], [57, -1], [58, 1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, -1], [70, 1], [71, 1], [72, -1], [73, -1], [74, -1], [75, -1], [76, 1], [77, 1], [78, 1], [79, 1], [80, 1], [81, 1], [82, 1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1], [89, -1], [90, -1], [91, -1], [92, 1], [93, -1], [94, -1], [95, 1], [96, 1], [97, 1], [98, -1], [99, -1], [100, 1]], Yes, zero_residue_nonresidue_partition_is_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0025/)

Legendre symbol · qf-mod43-legendre-classification-table · candidate





## 26. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=3, original a=1. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(3, 1, 1, [1, 2], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (3, 1, 1, [1, 2], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0026/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 27. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=5, original a=2. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(5, 2, 2, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (5, 2, 2, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0027/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 28. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=7, original a=3. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(7, 3, 3, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (7, 3, 3, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0028/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 29. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=11, original a=4. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(11, 4, 4, [2, 9], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (11, 4, 4, [2, 9], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0029/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 30. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=13, original a=5. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(13, 5, 5, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (13, 5, 5, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0030/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 31. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=17, original a=6. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(17, 6, 6, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (17, 6, 6, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0031/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 32. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=19, original a=7. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(19, 7, 7, [8, 11], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (19, 7, 7, [8, 11], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0032/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 33. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=23, original a=8. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(23, 8, 8, [10, 13], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (23, 8, 8, [10, 13], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0033/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 34. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=29, original a=9. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(29, 9, 9, [3, 26], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (29, 9, 9, [3, 26], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0034/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 35. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=31, original a=10. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(31, 10, 10, [14, 17], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (31, 10, 10, [14, 17], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0035/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 36. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=37, original a=11. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(37, 11, 11, [14, 23], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (37, 11, 11, [14, 23], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0036/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 37. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=41, original a=12. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(41, 12, 12, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (41, 12, 12, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0037/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 38. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=43, original a=13. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(43, 13, 13, [20, 23], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (43, 13, 13, [20, 23], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0038/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 39. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=47, original a=14. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(47, 14, 14, [22, 25], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (47, 14, 14, [22, 25], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0039/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 40. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=53, original a=15. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(53, 15, 15, [11, 42], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (53, 15, 15, [11, 42], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0040/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 41. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=59, original a=16. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(59, 16, 16, [4, 55], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (59, 16, 16, [4, 55], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0041/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 42. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=61, original a=17. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(61, 17, 17, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (61, 17, 17, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0042/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 43. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=67, original a=18. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(67, 18, 18, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (67, 18, 18, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0043/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 44. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=71, original a=19. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(71, 19, 19, [27, 44], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (71, 19, 19, [27, 44], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0044/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 45. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=73, original a=20. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(73, 20, 20, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (73, 20, 20, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0045/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 46. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=79, original a=21. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(79, 21, 21, [10, 69], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (79, 21, 21, [10, 69], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0046/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 47. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=83, original a=22. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(83, 22, 22, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (83, 22, 22, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0047/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 48. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=89, original a=23. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(89, 23, 23, [], -1, complete_square_table_has_no_root)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (89, 23, 23, [], -1, complete_square_table_has_no_root).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0048/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 49. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=97, original a=24. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(97, 24, 24, [11, 86], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (97, 24, 24, [11, 86], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0049/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 50. A Complete Root Table Distinguishes Symbol Plus One from Minus One

Use odd prime p=101, original a=25. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

View answer and explanation(101, 25, 25, [5, 96], 1, complete_nonzero_square_root_witness)

Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (101, 25, 25, [5, 96], 1, complete_nonzero_square_root_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0050/)

Legendre symbol · qf-mod43-legendre-residue-nonresidue-witness · candidate





## 51. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=3, original a=3. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(3, 3, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (3, 3, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0051/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 52. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=5, original a=10. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(5, 10, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (5, 10, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0052/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 53. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=7, original a=21. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(7, 21, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (7, 21, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0053/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 54. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=11, original a=44. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(11, 44, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (11, 44, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0054/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 55. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=13, original a=65. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(13, 65, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (13, 65, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0055/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 56. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=17, original a=102. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(17, 102, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (17, 102, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0056/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 57. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=19, original a=133. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(19, 133, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (19, 133, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0057/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 58. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=23, original a=184. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(23, 184, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (23, 184, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0058/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 59. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=29, original a=261. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(29, 261, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (29, 261, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0059/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 60. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=31, original a=310. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(31, 310, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (31, 310, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0060/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 61. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=37, original a=407. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(37, 407, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (37, 407, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0061/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 62. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=41, original a=492. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(41, 492, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (41, 492, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0062/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 63. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=43, original a=559. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(43, 559, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (43, 559, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0063/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 64. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=47, original a=658. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(47, 658, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (47, 658, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0064/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 65. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=53, original a=795. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(53, 795, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (53, 795, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0065/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 66. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=59, original a=944. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(59, 944, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (59, 944, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0066/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 67. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=61, original a=1037. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(61, 1037, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (61, 1037, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0067/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 68. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=67, original a=1206. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(67, 1206, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (67, 1206, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0068/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 69. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=71, original a=1349. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(71, 1349, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (71, 1349, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0069/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 70. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=73, original a=1460. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(73, 1460, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (73, 1460, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0070/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 71. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=79, original a=1659. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(79, 1659, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (79, 1659, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0071/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 72. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=83, original a=1826. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(83, 1826, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (83, 1826, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0072/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 73. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=89, original a=2047. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(89, 2047, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (89, 2047, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0073/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 74. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=97, original a=2328. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(97, 2328, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (97, 2328, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0074/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 75. Divisibility by p Gives Symbol Zero, Not Plus One

Use odd prime p=101, original a=2525. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).

The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only.

View answer and explanation(101, 2525, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (101, 2525, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0075/)

Legendre symbol · qf-mod43-legendre-zero-class · candidate





## 76. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0076/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 77. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=9, original a=2. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 9, 2, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 9, 2, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0077/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 78. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=7, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 7, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 7, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0078/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 79. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=4. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0079/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 80. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=15, original a=5. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 15, 5, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 15, 5, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0080/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 81. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=17, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 17, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 17, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0081/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 82. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=7. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0082/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 83. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=21, original a=8. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 21, 8, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 21, 8, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0083/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 84. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=29, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 29, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 29, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0084/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 85. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=10. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0085/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 86. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=27, original a=11. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 27, 11, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 27, 11, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0086/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 87. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=41, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 41, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 41, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0087/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 88. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=13. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0088/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 89. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=33, original a=14. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 33, 14, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 33, 14, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0089/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 90. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=53, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 53, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 53, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0090/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 91. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=16. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0091/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 92. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=39, original a=17. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 39, 17, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 39, 17, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0092/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 93. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=67, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 67, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 67, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0093/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 94. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=19. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0094/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 95. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=45, original a=20. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 45, 20, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 45, 20, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0095/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 96. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=79, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 79, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 79, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0096/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 97. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=22. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0097/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 98. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=composite_modulus, modulus=51, original a=23. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(composite_modulus, 51, 23, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 51, 23, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0098/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 99. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=premature_euler, modulus=97, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(premature_euler, 97, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 97, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0099/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate





## 100. Domain Extensions and Premature Methods Are Different Legendre Boundaries

Use mode=even_prime, modulus=2, original a=25. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain.

View answer and explanation(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261112-r06-legendre-symbol-0100/)

Legendre symbol · qf-mod43-legendre-domain-method-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r06-legendre-symbol-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261112-r06-legendre-symbol-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261112-r06-legendre-symbol-foundations/validation-report.json)