# Independent Square and Modular-Power Routes with Odd-Prime 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-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations · 2026-10-15 · 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. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=3, original a=1. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(3, 1, 1, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (3, 1, 1, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0001/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 2. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=5, original a=2. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(5, 2, 2, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (5, 2, 2, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0002/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 3. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=7, original a=3. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(7, 3, 3, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (7, 3, 3, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0003/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 4. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=11, original a=4. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(11, 4, 4, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (11, 4, 4, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0004/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 5. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=13, original a=5. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(13, 5, 5, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (13, 5, 5, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0005/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 6. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=17, original a=6. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(17, 6, 6, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (17, 6, 6, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0006/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 7. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=19, original a=7. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(19, 7, 7, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (19, 7, 7, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0007/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 8. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=23, original a=8. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(23, 8, 8, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (23, 8, 8, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0008/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 9. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=29, original a=9. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(29, 9, 9, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (29, 9, 9, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0009/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 10. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=31, original a=10. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(31, 10, 10, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (31, 10, 10, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0010/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 11. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=37, original a=11. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(37, 11, 11, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (37, 11, 11, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0011/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 12. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=41, original a=12. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(41, 12, 12, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (41, 12, 12, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0012/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 13. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=43, original a=13. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(43, 13, 13, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (43, 13, 13, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0013/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 14. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=47, original a=14. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(47, 14, 14, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (47, 14, 14, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0014/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 15. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=53, original a=15. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(53, 15, 15, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (53, 15, 15, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0015/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 16. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=59, original a=16. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(59, 16, 16, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (59, 16, 16, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0016/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 17. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=61, original a=17. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(61, 17, 17, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (61, 17, 17, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0017/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 18. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=67, original a=18. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(67, 18, 18, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (67, 18, 18, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0018/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 19. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=71, original a=19. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(71, 19, 19, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (71, 19, 19, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0019/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 20. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=73, original a=20. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(73, 20, 20, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (73, 20, 20, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0020/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 21. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=79, original a=21. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(79, 21, 21, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (79, 21, 21, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0021/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 22. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=83, original a=22. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(83, 22, 22, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (83, 22, 22, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0022/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 23. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=89, original a=23. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(89, 23, 23, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (89, 23, 23, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0023/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 24. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=97, original a=24. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(97, 24, 24, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (97, 24, 24, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0024/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 25. Qualify Odd Primality and the Normalized Nonzero Class First

Use prime=101, original a=25. Give (prime, a, normalized_class, odd_prime, nonzero_class, certificate).

View answer and explanation(101, 25, 25, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified)

Require a bounded odd prime and a mod p≠0 before this criterion. The Legendre symbol is already defined by square classification, not by assuming this theorem. The exact result, retaining the requested order, is (101, 25, 25, Yes, Yes, odd_prime_and_nonzero_modular_class_qualified).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0025/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-qualification · candidate





## 26. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (3, 1, [1, 2], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0026/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 27. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(5, 2, [], -1, 4, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (5, 2, [], -1, 4, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0027/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 28. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(7, 3, [], -1, 6, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (7, 3, [], -1, 6, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0028/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 29. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (11, 4, [2, 9], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0029/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 30. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(13, 5, [], -1, 12, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (13, 5, [], -1, 12, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0030/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 31. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(17, 6, [], -1, 16, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (17, 6, [], -1, 16, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0031/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 32. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (19, 7, [8, 11], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0032/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 33. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (23, 8, [10, 13], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0033/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 34. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (29, 9, [3, 26], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0034/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 35. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (31, 10, [14, 17], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0035/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 36. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (37, 11, [14, 23], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0036/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 37. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(41, 12, [], -1, 40, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (41, 12, [], -1, 40, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0037/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 38. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (43, 13, [20, 23], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0038/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 39. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (47, 14, [22, 25], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0039/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 40. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (53, 15, [11, 42], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0040/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 41. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (59, 16, [4, 55], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0041/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 42. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(61, 17, [], -1, 60, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (61, 17, [], -1, 60, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0042/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 43. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(67, 18, [], -1, 66, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (67, 18, [], -1, 66, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0043/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 44. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (71, 19, [27, 44], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0044/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 45. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(73, 20, [], -1, 72, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (73, 20, [], -1, 72, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0045/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 46. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (79, 21, [10, 69], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0046/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 47. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(83, 22, [], -1, 82, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (83, 22, [], -1, 82, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0047/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 48. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

View answer and explanation(89, 23, [], -1, 88, -1, Yes, square_table_and_repeated_squaring_are_independent_computations)

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (89, 23, [], -1, 88, -1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0048/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 49. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (97, 24, [11, 86], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0049/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 50. Compare a Square-Table Legendre Value with a Separately Computed Half Power

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

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

Normalize a and obtain all nonzero roots from the complete square table, giving Legendre±1 independently. Compute a^((p−1)/2) modulo p by repeated squaring, then interpret raw residue p−1 as signed−1. Compare signed values; one finite instance is not the general theorem proof. The exact result, retaining the requested order, is (101, 25, [5, 96], 1, 1, 1, Yes, square_table_and_repeated_squaring_are_independent_computations).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0050/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-independent-comparison · candidate





## 51. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=3, original a=1, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(3, 1, quadratic_residue, [1, 2], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (3, 1, quadratic_residue, [1, 2], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0051/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 52. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=5, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(5, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (5, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0052/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 53. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=7, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(7, 2, quadratic_residue, [3, 4], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (7, 2, quadratic_residue, [3, 4], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0053/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 54. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=11, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(11, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (11, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0054/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 55. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=13, original a=12, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(13, 12, quadratic_residue, [5, 8], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (13, 12, quadratic_residue, [5, 8], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0055/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 56. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=17, original a=3, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(17, 3, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (17, 3, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0056/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 57. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=19, original a=11, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(19, 11, quadratic_residue, [7, 12], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (19, 11, quadratic_residue, [7, 12], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0057/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 58. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=23, original a=5, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(23, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (23, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0058/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 59. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=29, original a=23, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(29, 23, quadratic_residue, [9, 20], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (29, 23, quadratic_residue, [9, 20], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0059/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 60. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=31, original a=3, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(31, 3, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (31, 3, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0060/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 61. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=37, original a=10, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(37, 10, quadratic_residue, [11, 26], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (37, 10, quadratic_residue, [11, 26], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0061/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 62. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=41, original a=3, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(41, 3, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (41, 3, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0062/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 63. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=43, original a=40, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(43, 40, quadratic_residue, [13, 30], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (43, 40, quadratic_residue, [13, 30], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0063/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 64. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=47, original a=5, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(47, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (47, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0064/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 65. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=53, original a=13, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(53, 13, quadratic_residue, [15, 38], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (53, 13, quadratic_residue, [15, 38], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0065/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 66. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=59, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(59, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (59, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0066/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 67. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=61, original a=45, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(61, 45, quadratic_residue, [17, 44], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (61, 45, quadratic_residue, [17, 44], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0067/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 68. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=67, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(67, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (67, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0068/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 69. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=71, original a=6, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(71, 6, quadratic_residue, [19, 52], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (71, 6, quadratic_residue, [19, 52], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0069/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 70. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=73, original a=5, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(73, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (73, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0070/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 71. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=79, original a=46, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(79, 46, quadratic_residue, [21, 58], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (79, 46, quadratic_residue, [21, 58], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0071/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 72. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=83, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(83, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (83, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0072/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 73. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=89, original a=84, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(89, 84, quadratic_residue, [23, 66], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (89, 84, quadratic_residue, [23, 66], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0073/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 74. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=97, original a=5, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(97, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (97, 5, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0074/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 75. Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Use odd prime p=101, original a=19, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

View answer and explanation(101, 19, quadratic_residue, [25, 76], 1, 1, Yes, residue_maps_to_plus_one)

Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (101, 19, quadratic_residue, [25, 76], 1, 1, Yes, residue_maps_to_plus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0075/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-residue-nonresidue-cases · candidate





## 76. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0076/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 77. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 9, 2, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0077/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 78. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=7, original a=21. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 7, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 7, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0078/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 79. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0079/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 80. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 15, 5, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0080/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 81. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=17, original a=102. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 17, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 17, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0081/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 82. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0082/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 83. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 21, 8, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0083/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 84. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=29, original a=261. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 29, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 29, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0084/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 85. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0085/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 86. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 27, 11, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0086/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 87. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=41, original a=492. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 41, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 41, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0087/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 88. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0088/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 89. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 33, 14, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0089/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 90. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=53, original a=795. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 53, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 53, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0090/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 91. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0091/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 92. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 39, 17, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0092/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 93. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=67, original a=1206. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 67, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 67, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0093/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 94. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0094/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 95. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 45, 20, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0095/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 96. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=79, original a=1659. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 79, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 79, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0096/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 97. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0097/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 98. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (composite_modulus, 51, 23, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0098/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 99. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

Use mode=zero_class, modulus=97, original a=2328. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

View answer and explanation(zero_class, 97, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (zero_class, 97, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0099/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate





## 100. Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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

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

Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0100/)

Euler criterion for quadratic residues · qf-mod44-eulercrit-domain-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations/validation-report.json)