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batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations · 2026-10-14 · 100 題
欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":5,"a":2}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(5, 2, 4, 1, 是)
Wave6根層核對=(5, 2, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":6,"a":5}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(6, 5, 2, 1, 是)
Wave6根層核對=(6, 5, 2, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":7,"a":4}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(7, 4, 6, 1, 是)
Wave6根層核對=(7, 4, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":8,"a":5}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(8, 5, 4, 1, 是)
Wave6根層核對=(8, 5, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":9,"a":7}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(9, 7, 6, 1, 是)
Wave6根層核對=(9, 7, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":10,"a":7}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(10, 7, 4, 1, 是)
Wave6根層核對=(10, 7, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":11,"a":8}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(11, 8, 10, 1, 是)
Wave6根層核對=(11, 8, 10, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":12,"a":11}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(12, 11, 4, 1, 是)
Wave6根層核對=(12, 11, 4, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":13,"a":10}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(13, 10, 12, 1, 是)
Wave6根層核對=(13, 10, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":14,"a":11}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(14, 11, 6, 1, 是)
Wave6根層核對=(14, 11, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":15,"a":13}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(15, 13, 8, 1, 是)
Wave6根層核對=(15, 13, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":16,"a":13}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(16, 13, 8, 1, 是)
Wave6根層核對=(16, 13, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":17,"a":14}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(17, 14, 16, 1, 是)
Wave6根層核對=(17, 14, 16, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":18,"a":17}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(18, 17, 6, 1, 是)
Wave6根層核對=(18, 17, 6, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":19,"a":16}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(19, 16, 18, 1, 是)
Wave6根層核對=(19, 16, 18, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":20,"a":17}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(20, 17, 8, 1, 是)
Wave6根層核對=(20, 17, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":21,"a":19}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(21, 19, 12, 1, 是)
Wave6根層核對=(21, 19, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":22,"a":19}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(22, 19, 10, 1, 是)
Wave6根層核對=(22, 19, 10, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":23,"a":20}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(23, 20, 22, 1, 是)
Wave6根層核對=(23, 20, 22, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":24,"a":23}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(24, 23, 8, 1, 是)
Wave6根層核對=(24, 23, 8, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":25,"a":22}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(25, 22, 20, 1, 是)
Wave6根層核對=(25, 22, 20, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":26,"a":23}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(26, 23, 12, 1, 是)
Wave6根層核對=(26, 23, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":27,"a":25}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(27, 25, 18, 1, 是)
Wave6根層核對=(27, 25, 18, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":28,"a":25}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(28, 25, 12, 1, 是)
Wave6根層核對=(28, 25, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={"n":29,"a":26}。請依 [n, normalized_base, totient, residue, congruent_to_one] 欄序回答。
(29, 26, 28, 1, 是)
Wave6根層核對=(29, 26, 28, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-congruence · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":5,"a":2}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 2, 3, 4], [1, 2, 3, 4], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 2, 3, 4], [1, 2, 3, 4], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":6,"a":5}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 5], [1, 5], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 5], [1, 5], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":7,"a":4}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":8,"a":5}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 5, 7], [1, 3, 5, 7], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 5, 7], [1, 3, 5, 7], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":9,"a":7}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 8], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":10,"a":7}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 7, 9], [1, 3, 7, 9], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 7, 9], [1, 3, 7, 9], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":11,"a":8}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":12,"a":11}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 5, 7, 11], [1, 5, 7, 11], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 5, 7, 11], [1, 5, 7, 11], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":13,"a":10}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":14,"a":11}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 5, 9, 11, 13], [1, 3, 5, 9, 11, 13], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":15,"a":13}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 2, 4, 7, 8, 11, 13, 14], [1, 2, 4, 7, 8, 11, 13, 14], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":16,"a":13}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 5, 7, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":17,"a":14}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([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], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([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], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":18,"a":17}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 5, 7, 11, 13, 17], [1, 5, 7, 11, 13, 17], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":19,"a":16}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([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], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([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], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":20,"a":17}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 7, 9, 11, 13, 17, 19], [1, 3, 7, 9, 11, 13, 17, 19], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":21,"a":19}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":22,"a":19}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 5, 7, 9, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 13, 15, 17, 19, 21], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":23,"a":20}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([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], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([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], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":24,"a":23}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 5, 7, 11, 13, 17, 19, 23], [1, 5, 7, 11, 13, 17, 19, 23], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":25,"a":22}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([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], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([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], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":26,"a":23}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":27,"a":25}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([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], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([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], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":28,"a":25}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], [1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-unit-permutation；資料={"n":29,"a":26}。請依 [units, sorted_mapped_units, permutation, certificate] 欄序回答。
([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], 是, multiplication_by_a_permutes_reduced_residues)
Wave6根層核對=([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], 是, multiplication_by_a_permutes_reduced_residues)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-unit-permutation · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":5,"a":-2}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-2, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-2, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":6,"a":-5}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-5, 1, 2, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-5, 1, 2, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":7,"a":-4}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-4, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-4, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":8,"a":-5}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-5, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-5, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":9,"a":-7}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-7, 2, 6, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-7, 2, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":10,"a":-7}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-7, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-7, 3, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":11,"a":-8}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-8, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-8, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":12,"a":-11}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-11, 1, 4, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-11, 1, 4, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":13,"a":-10}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-10, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-10, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":14,"a":-11}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-11, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-11, 3, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":15,"a":-13}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-13, 2, 8, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-13, 2, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":16,"a":-13}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-13, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-13, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":17,"a":-14}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-14, 3, 16, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-14, 3, 16, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":18,"a":-17}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-17, 1, 6, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-17, 1, 6, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":19,"a":-16}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-16, 3, 18, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-16, 3, 18, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":20,"a":-17}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-17, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-17, 3, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":21,"a":-19}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-19, 2, 12, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-19, 2, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":22,"a":-19}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-19, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-19, 3, 10, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":23,"a":-20}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-20, 3, 22, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-20, 3, 22, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":24,"a":-23}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-23, 1, 8, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-23, 1, 8, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":25,"a":-22}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-22, 3, 20, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-22, 3, 20, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":26,"a":-23}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-23, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-23, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":27,"a":-25}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-25, 2, 18, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-25, 2, 18, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":28,"a":-25}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-25, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-25, 3, 12, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":29,"a":-26}。請依 [original_base, normalized_base, totient, residue, congruent_to_one, certificate] 欄序回答。
(-26, 3, 28, 1, 是, negative_base_normalized_before_exact_modular_power)
Wave6根層核對=(-26, 3, 28, 1, 是, negative_base_normalized_before_exact_modular_power)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-negative-base-normalization · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":6,"a":6}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(6, 6, 6, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(6, 6, 6, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":7,"a":7}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(7, 7, 7, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(7, 7, 7, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":8,"a":8}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(8, 8, 8, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(8, 8, 8, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":9,"a":9}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(9, 9, 9, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(9, 9, 9, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":10,"a":10}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(10, 10, 10, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(10, 10, 10, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":11,"a":11}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(11, 11, 11, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(11, 11, 11, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":12,"a":12}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(12, 12, 12, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(12, 12, 12, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":13,"a":13}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(13, 13, 13, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(13, 13, 13, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":14,"a":14}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(14, 14, 14, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(14, 14, 14, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":15,"a":15}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(15, 15, 15, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(15, 15, 15, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":16,"a":16}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(16, 16, 16, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(16, 16, 16, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":17,"a":17}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(17, 17, 17, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(17, 17, 17, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":18,"a":18}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(18, 18, 18, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(18, 18, 18, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":19,"a":19}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(19, 19, 19, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(19, 19, 19, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":20,"a":20}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(20, 20, 20, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(20, 20, 20, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":21,"a":21}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(21, 21, 21, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(21, 21, 21, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":22,"a":22}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(22, 22, 22, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(22, 22, 22, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":23,"a":23}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(23, 23, 23, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(23, 23, 23, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":24,"a":24}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(24, 24, 24, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(24, 24, 24, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":25,"a":25}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(25, 25, 25, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(25, 25, 25, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":26,"a":26}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(26, 26, 26, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(26, 26, 26, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":27,"a":27}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(27, 27, 27, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(27, 27, 27, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":28,"a":28}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(28, 28, 28, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(28, 28, 28, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":29,"a":29}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(29, 29, 29, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(29, 29, 29, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate
n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-nonunit-boundary；資料={"n":30,"a":30}。請依 [n, original_base, gcd, theorem_applicable, residue, boundary] 欄序回答。
(30, 30, 30, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)
Wave6根層核對=(30, 30, 30, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。
Euler定理 · qf-mod38-eulerthm-nonunit-boundary · candidate