第 20078 題：Euler定理

原題與已知條件

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 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 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 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-euler-theorem-number-theory

逐步推導

source operation=eulerthm-unit-permutation;original snapshot and mathematics retained

explicit ordered source output fields=[units, sorted_mapped_units, permutation, certificate]

Euler ORIGINAL signed base normalization exact quotient := div(4, 7)

Euler ORIGINAL signed base normalization exact quotient = 4/7

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 7)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(4, 0)

Euler ORIGINAL signed base normalization normalized remainder = 4

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=4;modulus=7;exact ratio=4/7;floor quotient=0

value=q*n+remainder;normalized residue=4 in[0,7);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[4, 7];nonnegative initial pair=[4, 7]

ALL exact Euclidean rows[left,right,q,r]=[[4, 7, 0, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 7];nonnegative initial pair=[0, 7]

ALL exact Euclidean rows[left,right,q,r]=[[0, 7, 0, 0]];terminal gcd=7

Euler residue representative1:ORIGINAL gcd inputs=[1, 7];nonnegative initial pair=[1, 7]

ALL exact Euclidean rows[left,right,q,r]=[[1, 7, 0, 1], [7, 1, 7, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 7];nonnegative initial pair=[2, 7]

ALL exact Euclidean rows[left,right,q,r]=[[2, 7, 0, 2], [7, 2, 3, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 7];nonnegative initial pair=[3, 7]

ALL exact Euclidean rows[left,right,q,r]=[[3, 7, 0, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 7];nonnegative initial pair=[4, 7]

ALL exact Euclidean rows[left,right,q,r]=[[4, 7, 0, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 7];nonnegative initial pair=[5, 7]

ALL exact Euclidean rows[left,right,q,r]=[[5, 7, 0, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 7];nonnegative initial pair=[6, 7]

ALL exact Euclidean rows[left,right,q,r]=[[6, 7, 0, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal gcd=1

Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1)

Euler complete reduced residue count phi = 6

COMPLETE original residue carrier0..6;ALL [representative,gcd,unit] rows=[[0, 7, 否], [1, 1, 是], [2, 1, 是], [3, 1, 是], [4, 1, 是], [5, 1, 是], [6, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6];phi(n)=6;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Euler ACTUAL base-unit inverse product := mul(4, 2)

Euler ACTUAL base-unit inverse product = 8

Euler ACTUAL inverse check exact quotient := div(8, 7)

Euler ACTUAL inverse check exact quotient = 8/7

Euler ACTUAL inverse check floor quotient multiple := mul(1, 7)

Euler ACTUAL inverse check floor quotient multiple = 7

Euler ACTUAL inverse check normalized remainder := sub(8, 7)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=8;modulus=7;exact ratio=8/7;floor quotient=1

value=q*n+remainder;normalized residue=1 in[0,7);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=7;gcd(original a=4,n)=1;normalized base=4;ACTUAL inverse=2;COMPLETE U(n)=[1, 2, 3, 4, 5, 6]

for ANY r in U(n),a*r remains a unit;actual inverse makes multiplication injective;finite self-injection on U(n) is surjective,so a*U(n)=U(n)

Euler ORIGINAL a times unit representative1 := mul(4, 1)

Euler ORIGINAL a times unit representative1 = 4

Euler unit representative1 raw image exact quotient := div(4, 7)

Euler unit representative1 raw image exact quotient = 4/7

Euler unit representative1 raw image floor quotient multiple := mul(0, 7)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(4, 0)

Euler unit representative1 raw image normalized remainder = 4

Euler unit representative1 raw image:ORIGINAL signed integer=4;modulus=7;exact ratio=4/7;floor quotient=0

value=q*n+remainder;normalized residue=4 in[0,7);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(4, 2)

Euler ORIGINAL a times unit representative2 = 8

Euler unit representative2 raw image exact quotient := div(8, 7)

Euler unit representative2 raw image exact quotient = 8/7

Euler unit representative2 raw image floor quotient multiple := mul(1, 7)

Euler unit representative2 raw image floor quotient multiple = 7

Euler unit representative2 raw image normalized remainder := sub(8, 7)

Euler unit representative2 raw image normalized remainder = 1

Euler unit representative2 raw image:ORIGINAL signed integer=8;modulus=7;exact ratio=8/7;floor quotient=1

value=q*n+remainder;normalized residue=1 in[0,7);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative3 := mul(4, 3)

Euler ORIGINAL a times unit representative3 = 12

Euler unit representative3 raw image exact quotient := div(12, 7)

Euler unit representative3 raw image exact quotient = 12/7

Euler unit representative3 raw image floor quotient multiple := mul(1, 7)

Euler unit representative3 raw image floor quotient multiple = 7

Euler unit representative3 raw image normalized remainder := sub(12, 7)

Euler unit representative3 raw image normalized remainder = 5

Euler unit representative3 raw image:ORIGINAL signed integer=12;modulus=7;exact ratio=12/7;floor quotient=1

value=q*n+remainder;normalized residue=5 in[0,7);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative4 := mul(4, 4)

Euler ORIGINAL a times unit representative4 = 16

Euler unit representative4 raw image exact quotient := div(16, 7)

Euler unit representative4 raw image exact quotient = 16/7

Euler unit representative4 raw image floor quotient multiple := mul(2, 7)

Euler unit representative4 raw image floor quotient multiple = 14

Euler unit representative4 raw image normalized remainder := sub(16, 14)

Euler unit representative4 raw image normalized remainder = 2

Euler unit representative4 raw image:ORIGINAL signed integer=16;modulus=7;exact ratio=16/7;floor quotient=2

value=q*n+remainder;normalized residue=2 in[0,7);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(4, 5)

Euler ORIGINAL a times unit representative5 = 20

Euler unit representative5 raw image exact quotient := div(20, 7)

Euler unit representative5 raw image exact quotient = 20/7

Euler unit representative5 raw image floor quotient multiple := mul(2, 7)

Euler unit representative5 raw image floor quotient multiple = 14

Euler unit representative5 raw image normalized remainder := sub(20, 14)

Euler unit representative5 raw image normalized remainder = 6

Euler unit representative5 raw image:ORIGINAL signed integer=20;modulus=7;exact ratio=20/7;floor quotient=2

value=q*n+remainder;normalized residue=6 in[0,7);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative6 := mul(4, 6)

Euler ORIGINAL a times unit representative6 = 24

Euler unit representative6 raw image exact quotient := div(24, 7)

Euler unit representative6 raw image exact quotient = 24/7

Euler unit representative6 raw image floor quotient multiple := mul(3, 7)

Euler unit representative6 raw image floor quotient multiple = 21

Euler unit representative6 raw image normalized remainder := sub(24, 21)

Euler unit representative6 raw image normalized remainder = 3

Euler unit representative6 raw image:ORIGINAL signed integer=24;modulus=7;exact ratio=24/7;floor quotient=3

value=q*n+remainder;normalized residue=3 in[0,7);same congruence class,NOT absolute-value replacement

COMPLETE original increasing U(n)=[1, 2, 3, 4, 5, 6];actual images in ORIGINAL unit order=[4, 1, 5, 2, 6, 3]

SOURCE output sorts the mapped list numerically=[1, 2, 3, 4, 5, 6];full-list equality=是;not a claim each point is fixed

逐步反向核對

• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S050 · premise_and_result_audit — 以精確有理數重算 Euler unit representative6 raw image normalized remainder，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Euler unit representative6 raw image floor quotient multiple，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Euler unit representative6 raw image exact quotient，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative6，不以小數近似替代。
• S046 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S045 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image normalized remainder，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image floor quotient multiple，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Euler unit representative5 raw image exact quotient，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative5，不以小數近似替代。
• S041 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S040 · premise_and_result_audit — 以精確有理數重算 Euler unit representative4 raw image normalized remainder，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Euler unit representative4 raw image floor quotient multiple，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Euler unit representative4 raw image exact quotient，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative4，不以小數近似替代。
• S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S035 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image normalized remainder，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image floor quotient multiple，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Euler unit representative3 raw image exact quotient，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative3，不以小數近似替代。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 Euler unit representative2 raw image normalized remainder，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Euler unit representative2 raw image floor quotient multiple，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Euler unit representative2 raw image exact quotient，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative2，不以小數近似替代。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image normalized remainder，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image floor quotient multiple，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Euler unit representative1 raw image exact quotient，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL a times unit representative1，不以小數近似替代。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check floor quotient multiple，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check exact quotient，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL base-unit inverse product，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization normalized remainder，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization floor quotient multiple，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization exact quotient，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], 是, multiplication_by_a_permutes_reduced_residues)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。