第 20056 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-nonunit-boundary；資料={"n":11,"a":11}。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-nonunit-boundary；資料={"n":11,"a":11}。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-nonunit-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[n, original_base, gcd, theorem_applicable, residue, boundary]

Euler ORIGINAL signed base normalization exact quotient := div(11, 11)

Euler ORIGINAL signed base normalization exact quotient = 1

Euler ORIGINAL signed base normalization floor quotient multiple := mul(1, 11)

Euler ORIGINAL signed base normalization floor quotient multiple = 11

Euler ORIGINAL signed base normalization normalized remainder := sub(11, 11)

Euler ORIGINAL signed base normalization normalized remainder = 0

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=11;modulus=11;exact ratio=1;floor quotient=1

value=q*n+remainder;normalized residue=0 in[0,11);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[11, 11];nonnegative initial pair=[11, 11]

ALL exact Euclidean rows[left,right,q,r]=[[11, 11, 1, 0]];terminal gcd=11

Euler residue representative0:ORIGINAL gcd inputs=[0, 11];nonnegative initial pair=[0, 11]

ALL exact Euclidean rows[left,right,q,r]=[[0, 11, 0, 0]];terminal gcd=11

Euler residue representative1:ORIGINAL gcd inputs=[1, 11];nonnegative initial pair=[1, 11]

ALL exact Euclidean rows[left,right,q,r]=[[1, 11, 0, 1], [11, 1, 11, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 11];nonnegative initial pair=[2, 11]

ALL exact Euclidean rows[left,right,q,r]=[[2, 11, 0, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 11];nonnegative initial pair=[3, 11]

ALL exact Euclidean rows[left,right,q,r]=[[3, 11, 0, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 11];nonnegative initial pair=[4, 11]

ALL exact Euclidean rows[left,right,q,r]=[[4, 11, 0, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 11];nonnegative initial pair=[5, 11]

ALL exact Euclidean rows[left,right,q,r]=[[5, 11, 0, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 11];nonnegative initial pair=[6, 11]

ALL exact Euclidean rows[left,right,q,r]=[[6, 11, 0, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative7:ORIGINAL gcd inputs=[7, 11];nonnegative initial pair=[7, 11]

ALL exact Euclidean rows[left,right,q,r]=[[7, 11, 0, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 11];nonnegative initial pair=[8, 11]

ALL exact Euclidean rows[left,right,q,r]=[[8, 11, 0, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 11];nonnegative initial pair=[9, 11]

ALL exact Euclidean rows[left,right,q,r]=[[9, 11, 0, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 11];nonnegative initial pair=[10, 11]

ALL exact Euclidean rows[left,right,q,r]=[[10, 11, 0, 10], [11, 10, 1, 1], [10, 1, 10, 0]];terminal gcd=1

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

Euler complete reduced residue count phi = 10

COMPLETE original residue carrier0..10;ALL [representative,gcd,unit] rows=[[0, 11, 否], [1, 1, 是], [2, 1, 是], [3, 1, 是], [4, 1, 是], [5, 1, 是], [6, 1, 是], [7, 1, 是], [8, 1, 是], [9, 1, 是], [10, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10];phi(n)=10;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

ORIGINAL normalized base=0;COMPLETE totient exponent=10;modulus=11

initialize accumulator1,current_base=base,remaining=phi;invariant accumulator*current_base^remaining congruent to the original totient power

Euler power bit0 base square := mul(0, 0)

Euler power bit0 base square = 0

Euler power bit0 squared-base reduction exact quotient := div(0, 11)

Euler power bit0 squared-base reduction exact quotient = 0

Euler power bit0 squared-base reduction floor quotient multiple := mul(0, 11)

Euler power bit0 squared-base reduction floor quotient multiple = 0

Euler power bit0 squared-base reduction normalized remainder := sub(0, 0)

Euler power bit0 squared-base reduction normalized remainder = 0

Euler power bit0 squared-base reduction:ORIGINAL signed integer=0;modulus=11;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,11);same congruence class,NOT absolute-value replacement

step=0;previous[accumulator,base,exponent]=[1, 0, 10];low bit=0

next[accumulator,base,exponent]=[1, 0, 5];split exponent=2*q+bit preserves original-power invariant

Euler power bit1 accumulator product := mul(1, 0)

Euler power bit1 accumulator product = 0

Euler power bit1 accumulator reduction exact quotient := div(0, 11)

Euler power bit1 accumulator reduction exact quotient = 0

Euler power bit1 accumulator reduction floor quotient multiple := mul(0, 11)

Euler power bit1 accumulator reduction floor quotient multiple = 0

Euler power bit1 accumulator reduction normalized remainder := sub(0, 0)

Euler power bit1 accumulator reduction normalized remainder = 0

Euler power bit1 accumulator reduction:ORIGINAL signed integer=0;modulus=11;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,11);same congruence class,NOT absolute-value replacement

Euler power bit1 base square := mul(0, 0)

Euler power bit1 base square = 0

Euler power bit1 squared-base reduction exact quotient := div(0, 11)

Euler power bit1 squared-base reduction exact quotient = 0

Euler power bit1 squared-base reduction floor quotient multiple := mul(0, 11)

Euler power bit1 squared-base reduction floor quotient multiple = 0

Euler power bit1 squared-base reduction normalized remainder := sub(0, 0)

Euler power bit1 squared-base reduction normalized remainder = 0

Euler power bit1 squared-base reduction:ORIGINAL signed integer=0;modulus=11;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,11);same congruence class,NOT absolute-value replacement

step=1;previous[accumulator,base,exponent]=[1, 0, 5];low bit=1

next[accumulator,base,exponent]=[0, 0, 2];split exponent=2*q+bit preserves original-power invariant

Euler power bit2 base square := mul(0, 0)

Euler power bit2 base square = 0

Euler power bit2 squared-base reduction exact quotient := div(0, 11)

Euler power bit2 squared-base reduction exact quotient = 0

Euler power bit2 squared-base reduction floor quotient multiple := mul(0, 11)

Euler power bit2 squared-base reduction floor quotient multiple = 0

Euler power bit2 squared-base reduction normalized remainder := sub(0, 0)

Euler power bit2 squared-base reduction normalized remainder = 0

Euler power bit2 squared-base reduction:ORIGINAL signed integer=0;modulus=11;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,11);same congruence class,NOT absolute-value replacement

step=2;previous[accumulator,base,exponent]=[0, 0, 2];low bit=0

next[accumulator,base,exponent]=[0, 0, 1];split exponent=2*q+bit preserves original-power invariant

Euler power bit3 accumulator product := mul(0, 0)

Euler power bit3 accumulator product = 0

Euler power bit3 accumulator reduction exact quotient := div(0, 11)

Euler power bit3 accumulator reduction exact quotient = 0

Euler power bit3 accumulator reduction floor quotient multiple := mul(0, 11)

Euler power bit3 accumulator reduction floor quotient multiple = 0

Euler power bit3 accumulator reduction normalized remainder := sub(0, 0)

Euler power bit3 accumulator reduction normalized remainder = 0

Euler power bit3 accumulator reduction:ORIGINAL signed integer=0;modulus=11;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,11);same congruence class,NOT absolute-value replacement

Euler power bit3 base square := mul(0, 0)

Euler power bit3 base square = 0

Euler power bit3 squared-base reduction exact quotient := div(0, 11)

Euler power bit3 squared-base reduction exact quotient = 0

Euler power bit3 squared-base reduction floor quotient multiple := mul(0, 11)

Euler power bit3 squared-base reduction floor quotient multiple = 0

Euler power bit3 squared-base reduction normalized remainder := sub(0, 0)

Euler power bit3 squared-base reduction normalized remainder = 0

Euler power bit3 squared-base reduction:ORIGINAL signed integer=0;modulus=11;exact ratio=0;floor quotient=0

value=q*n+remainder;normalized residue=0 in[0,11);same congruence class,NOT absolute-value replacement

step=3;previous[accumulator,base,exponent]=[0, 0, 1];low bit=1

next[accumulator,base,exponent]=[0, 0, 0];split exponent=2*q+bit preserves original-power invariant

remaining exponent0;final normalized accumulator=0

EXACT original normalized base^10 modulo11=0

ACTUAL original n=11,a=11;gcd=11>1;phi(n)=10;computed totient-power residue=0

Euler theorem applicability=否 because gcd1 premise fails;retain ACTUAL residue;do NOT cancel a nonunit or infer prime/composite status from this marker

逐步反向核對

• S056 · 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。
• S055 · 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。
• S054 · 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。
• S053 · 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。
• S052 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction normalized remainder，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction floor quotient multiple，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction exact quotient，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 base square，不以小數近似替代。
• S048 · 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。
• S047 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 accumulator reduction normalized remainder，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 accumulator reduction floor quotient multiple，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 accumulator reduction exact quotient，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 accumulator product，不以小數近似替代。
• S043 · 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。
• S042 · 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。
• S041 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction normalized remainder，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction floor quotient multiple，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction exact quotient，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 base square，不以小數近似替代。
• S037 · 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。
• 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 power bit1 squared-base reduction normalized remainder，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 squared-base reduction 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 power bit1 squared-base reduction exact quotient，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 base square，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 power bit1 accumulator reduction normalized remainder，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 accumulator reduction 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 power bit1 accumulator reduction exact quotient，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 accumulator product，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction normalized remainder，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction floor quotient multiple，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction exact quotient，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 base square，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · 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。
• S017 · 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。
• S016 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• 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

(11, 11, 11, 否, 0, Euler_theorem_not_applicable_and_no_primality_converse)

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。