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