Unit Residues, Exact Totient Powers and Signed Base Normalization
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batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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1. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=5≥2 and original integer base a=2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(5, 2, 4, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (5, 2, 4, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
2. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=6≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(6, 5, 2, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (6, 5, 2, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
3. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=7≥2 and original integer base a=4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(7, 4, 6, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (7, 4, 6, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
4. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=8≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(8, 5, 4, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (8, 5, 4, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
5. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=9≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(9, 7, 6, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (9, 7, 6, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
6. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(10, 7, 4, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (10, 7, 4, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
7. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(11, 8, 10, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (11, 8, 10, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
8. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=12≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(12, 11, 4, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (12, 11, 4, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
9. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(13, 10, 12, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (13, 10, 12, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
10. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=14≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(14, 11, 6, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (14, 11, 6, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
11. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=15≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(15, 13, 8, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (15, 13, 8, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
12. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=16≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(16, 13, 8, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (16, 13, 8, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
13. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=17≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(17, 14, 16, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (17, 14, 16, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
14. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=18≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(18, 17, 6, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (18, 17, 6, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
15. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=19≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(19, 16, 18, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (19, 16, 18, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
16. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=20≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(20, 17, 8, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (20, 17, 8, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
17. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=21≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(21, 19, 12, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (21, 19, 12, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
18. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=22≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(22, 19, 10, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (22, 19, 10, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
19. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=23≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(23, 20, 22, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (23, 20, 22, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
20. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=24≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(24, 23, 8, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (24, 23, 8, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
21. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=25≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(25, 22, 20, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (25, 22, 20, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
22. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(26, 23, 12, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (26, 23, 12, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
23. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=27≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(27, 25, 18, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (27, 25, 18, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
24. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=28≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(28, 25, 12, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (28, 25, 12, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
25. Euler Congruence Requires a Unit Residue, Not a Prime Modulus
Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).
View answer and explanation
(29, 26, 28, 1, Yes)
Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (29, 26, 28, 1, Yes).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-congruence · candidate
26. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=5≥2 and original integer base a=2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4], [1, 2, 3, 4], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4], [1, 2, 3, 4], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
27. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=6≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 5], [1, 5], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5], [1, 5], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
28. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=7≥2 and original integer base a=4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
29. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=8≥2 and original integer base a=5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 5, 7], [1, 3, 5, 7], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1, 3, 5, 7], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
30. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=9≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
31. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=10≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 7, 9], [1, 3, 7, 9], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 7, 9], [1, 3, 7, 9], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
32. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
33. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=12≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 5, 7, 11], [1, 5, 7, 11], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11], [1, 5, 7, 11], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
34. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=13≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
35. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=14≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
36. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=15≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
37. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=16≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
38. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=17≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
39. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=18≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
40. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=19≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
41. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=20≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
42. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=21≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
43. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=22≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
44. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=23≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
45. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=24≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
46. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=25≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
47. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
48. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=27≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], [1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
49. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=28≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
50. Multiplication by a Unit Permutes the Complete Reduced Residues
Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).
View answer and explanation
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], Yes, multiplication_by_a_permutes_reduced_residues)
List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], Yes, multiplication_by_a_permutes_reduced_residues).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-unit-permutation · candidate
51. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=5≥2 and original integer base a=-2. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-2, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-2, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
52. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=6≥2 and original integer base a=-5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-5, 1, 2, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-5, 1, 2, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
53. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=7≥2 and original integer base a=-4. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-4, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-4, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
54. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=8≥2 and original integer base a=-5. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-5, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-5, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
55. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=9≥2 and original integer base a=-7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-7, 2, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-7, 2, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
56. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=10≥2 and original integer base a=-7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-7, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-7, 3, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
57. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=11≥2 and original integer base a=-8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-8, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-8, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
58. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=12≥2 and original integer base a=-11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-11, 1, 4, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-11, 1, 4, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
59. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=13≥2 and original integer base a=-10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-10, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-10, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
60. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=14≥2 and original integer base a=-11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-11, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-11, 3, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
61. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=15≥2 and original integer base a=-13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-13, 2, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-13, 2, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
62. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=16≥2 and original integer base a=-13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-13, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-13, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
63. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=17≥2 and original integer base a=-14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-14, 3, 16, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-14, 3, 16, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
64. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=18≥2 and original integer base a=-17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-17, 1, 6, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-17, 1, 6, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
65. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=19≥2 and original integer base a=-16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-16, 3, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-16, 3, 18, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
66. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=20≥2 and original integer base a=-17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-17, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-17, 3, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
67. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=21≥2 and original integer base a=-19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-19, 2, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-19, 2, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
68. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=22≥2 and original integer base a=-19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-19, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-19, 3, 10, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
69. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=23≥2 and original integer base a=-20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-20, 3, 22, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-20, 3, 22, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
70. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=24≥2 and original integer base a=-23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-23, 1, 8, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-23, 1, 8, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
71. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=25≥2 and original integer base a=-22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-22, 3, 20, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-22, 3, 20, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
72. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=26≥2 and original integer base a=-23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-23, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-23, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
73. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=27≥2 and original integer base a=-25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-25, 2, 18, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-25, 2, 18, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
74. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=28≥2 and original integer base a=-25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-25, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-25, 3, 12, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
75. Normalize the Signed Base before Exact Modular Exponentiation
Use integer modulus n=29≥2 and original integer base a=-26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (original_base, normalized_base, totient, residue, congruent_to_one, certificate).
View answer and explanation
(-26, 3, 28, 1, Yes, negative_base_normalized_before_exact_modular_power)
Require a unit, retain the original signed base separately from its normalized residue, then compute the totient power. Negative bases are not replaced by their absolute values. The exact result, retaining the requested order, is (-26, 3, 28, 1, Yes, negative_base_normalized_before_exact_modular_power).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-negative-base-normalization · candidate
76. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=6≥2 and original integer base a=6. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(6, 6, 6, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (6, 6, 6, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
77. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=7≥2 and original integer base a=7. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(7, 7, 7, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (7, 7, 7, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
78. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=8≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(8, 8, 8, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (8, 8, 8, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
79. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=9≥2 and original integer base a=9. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(9, 9, 9, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (9, 9, 9, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
80. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=10≥2 and original integer base a=10. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(10, 10, 10, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (10, 10, 10, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
81. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=11≥2 and original integer base a=11. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(11, 11, 11, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (11, 11, 11, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
82. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=12≥2 and original integer base a=12. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(12, 12, 12, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (12, 12, 12, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
83. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=13≥2 and original integer base a=13. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(13, 13, 13, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (13, 13, 13, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
84. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=14≥2 and original integer base a=14. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(14, 14, 14, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (14, 14, 14, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
85. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=15≥2 and original integer base a=15. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(15, 15, 15, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (15, 15, 15, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
86. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=16≥2 and original integer base a=16. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(16, 16, 16, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (16, 16, 16, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
87. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=17≥2 and original integer base a=17. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(17, 17, 17, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (17, 17, 17, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
88. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=18≥2 and original integer base a=18. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(18, 18, 18, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (18, 18, 18, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
89. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=19≥2 and original integer base a=19. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(19, 19, 19, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (19, 19, 19, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
90. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=20≥2 and original integer base a=20. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(20, 20, 20, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (20, 20, 20, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
91. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=21≥2 and original integer base a=21. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(21, 21, 21, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (21, 21, 21, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
92. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=22≥2 and original integer base a=22. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(22, 22, 22, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (22, 22, 22, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
93. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=23≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(23, 23, 23, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (23, 23, 23, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
94. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=24≥2 and original integer base a=24. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(24, 24, 24, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (24, 24, 24, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
95. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=25≥2 and original integer base a=25. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(25, 25, 25, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (25, 25, 25, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
96. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=26≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(26, 26, 26, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (26, 26, 26, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
97. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=27≥2 and original integer base a=27. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(27, 27, 27, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (27, 27, 27, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
98. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=28≥2 and original integer base a=28. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(28, 28, 28, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (28, 28, 28, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
99. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=29≥2 and original integer base a=29. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(29, 29, 29, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (29, 29, 29, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
100. A Nonunit Fails Euler Applicability, Not a Primality Test
Use integer modulus n=30≥2 and original integer base a=30. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, original_base, gcd, theorem_applicable, residue, boundary).
View answer and explanation
(30, 30, 30, No, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Require gcd(a,n)>1, report that gcd and No applicability, and keep the actual totient-power residue. Do not cancel a nonunit or interpret a modular calculation as a primality certificate. The exact result, retaining the requested order, is (30, 30, 30, No, 0, Euler_theorem_not_applicable_and_no_primality_converse).
Reviewed step-by-step derivation
Euler's theorem in number theory · qf-mod38-eulerthm-nonunit-boundary · candidate
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