第 19874 題：Fermat小定理

原題與已知條件

exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={"modulus":27,"a":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={"modulus":27,"a":-25}。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-fermat-little-theorem

逐步推導

source operation=flt-hypothesis-audit;original snapshot and mathematics retained

explicit ordered source output fields=[modulus, original_base, prime, coprime, coprime_form_applicable, power_modulus_minus_1, boundary]

ACTUAL integer modulus=27;complete trial range2..floor_sqrt=5

checked rows[divisor,quotient,remainder]=[[2, 13, 1], [3, 9, 0]];first exact factor=3;prime=否

original signed base=-25;positive modulus=27;use nonnegative gcd inputs=[25, 27]

ALL Euclidean rows[left,right,q,r]=[[25, 27, 0, 25], [27, 25, 1, 2], [25, 2, 12, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-25, 27)

Fermat original signed base exact quotient = -25/27

Fermat original signed base floor quotient multiple := mul(-1, 27)

Fermat original signed base floor quotient multiple = -27

Fermat original signed base normalized remainder := sub(-25, -27)

Fermat original signed base normalized remainder = 2

Fermat original signed base:signed integer value=-25;positive modulus=27;exact quotient=-25/27;floor q=-1

value=q*modulus+remainder;normalized remainder=2 in[0,27)

Fermat original modular exponent:original normalized base=2;original exponent=26;modulus=27

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

Fermat original modular exponent bit0 current base square := mul(2, 2)

Fermat original modular exponent bit0 current base square = 4

Fermat original modular exponent bit0 squared base reduction exact quotient := div(4, 27)

Fermat original modular exponent bit0 squared base reduction exact quotient = 4/27

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(0, 27)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(4, 0)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 4

Fermat original modular exponent bit0 squared base reduction:signed integer value=4;positive modulus=27;exact quotient=4/27;floor q=0

value=q*modulus+remainder;normalized remainder=4 in[0,27)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 2, 26];low bit=0

next[accumulator,base,exponent]=[1, 4, 13];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 4)

Fermat original modular exponent bit1 accumulator product = 4

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(4, 27)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 4/27

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 27)

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(4, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 4

Fermat original modular exponent bit1 accumulator reduction:signed integer value=4;positive modulus=27;exact quotient=4/27;floor q=0

value=q*modulus+remainder;normalized remainder=4 in[0,27)

Fermat original modular exponent bit1 current base square := mul(4, 4)

Fermat original modular exponent bit1 current base square = 16

Fermat original modular exponent bit1 squared base reduction exact quotient := div(16, 27)

Fermat original modular exponent bit1 squared base reduction exact quotient = 16/27

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 27)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(16, 0)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 16

Fermat original modular exponent bit1 squared base reduction:signed integer value=16;positive modulus=27;exact quotient=16/27;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,27)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 4, 13];low bit=1

next[accumulator,base,exponent]=[4, 16, 6];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(16, 16)

Fermat original modular exponent bit2 current base square = 256

Fermat original modular exponent bit2 squared base reduction exact quotient := div(256, 27)

Fermat original modular exponent bit2 squared base reduction exact quotient = 256/27

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(9, 27)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 243

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(256, 243)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 13

Fermat original modular exponent bit2 squared base reduction:signed integer value=256;positive modulus=27;exact quotient=256/27;floor q=9

value=q*modulus+remainder;normalized remainder=13 in[0,27)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[4, 16, 6];low bit=0

next[accumulator,base,exponent]=[4, 13, 3];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 accumulator product := mul(4, 13)

Fermat original modular exponent bit3 accumulator product = 52

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(52, 27)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 52/27

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(1, 27)

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple = 27

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(52, 27)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 25

Fermat original modular exponent bit3 accumulator reduction:signed integer value=52;positive modulus=27;exact quotient=52/27;floor q=1

value=q*modulus+remainder;normalized remainder=25 in[0,27)

Fermat original modular exponent bit3 current base square := mul(13, 13)

Fermat original modular exponent bit3 current base square = 169

Fermat original modular exponent bit3 squared base reduction exact quotient := div(169, 27)

Fermat original modular exponent bit3 squared base reduction exact quotient = 169/27

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(6, 27)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 162

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(169, 162)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 7

Fermat original modular exponent bit3 squared base reduction:signed integer value=169;positive modulus=27;exact quotient=169/27;floor q=6

value=q*modulus+remainder;normalized remainder=7 in[0,27)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[4, 13, 3];low bit=1

next[accumulator,base,exponent]=[25, 7, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(25, 7)

Fermat original modular exponent bit4 accumulator product = 175

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(175, 27)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 175/27

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(6, 27)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 162

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(175, 162)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 13

Fermat original modular exponent bit4 accumulator reduction:signed integer value=175;positive modulus=27;exact quotient=175/27;floor q=6

value=q*modulus+remainder;normalized remainder=13 in[0,27)

Fermat original modular exponent bit4 current base square := mul(7, 7)

Fermat original modular exponent bit4 current base square = 49

Fermat original modular exponent bit4 squared base reduction exact quotient := div(49, 27)

Fermat original modular exponent bit4 squared base reduction exact quotient = 49/27

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(1, 27)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 27

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(49, 27)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 22

Fermat original modular exponent bit4 squared base reduction:signed integer value=49;positive modulus=27;exact quotient=49/27;floor q=1

value=q*modulus+remainder;normalized remainder=22 in[0,27)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[25, 7, 1];low bit=1

next[accumulator,base,exponent]=[13, 22, 0];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent:remaining exponent0;final normalized accumulator=13

EXACT original base^26 modulo27=13

ACTUAL modulus=27;ORIGINAL signed base=-25;prime=否;coprime=是;actual power residue=13

prime-coprime applicability=否;failed hypotheses do NOT force actual residue!=1;one successful residue does NOT prove prime

逐步反向核對

• 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 — 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 — 以精確有理數重算 Fermat original modular exponent bit4 squared base reduction normalized remainder，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 squared base reduction floor quotient multiple，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 squared base reduction exact quotient，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 current base square，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 accumulator reduction normalized remainder，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 accumulator reduction floor quotient multiple，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 accumulator reduction exact quotient，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 accumulator product，不以小數近似替代。
• 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 — 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 Fermat original modular exponent bit3 squared base reduction normalized remainder，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction floor quotient multiple，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction exact quotient，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 current base square，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 Fermat original modular exponent bit3 accumulator reduction normalized remainder，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 accumulator 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 Fermat original modular exponent bit3 accumulator reduction exact quotient，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 accumulator product，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction normalized remainder，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction floor quotient multiple，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction exact quotient，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 current base square，不以小數近似替代。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 squared base reduction normalized remainder，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 squared base reduction floor quotient multiple，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 squared base reduction exact quotient，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 current base square，不以小數近似替代。
• 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 Fermat original modular exponent bit1 accumulator reduction normalized remainder，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 accumulator reduction floor quotient multiple，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 accumulator reduction exact quotient，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 accumulator product，不以小數近似替代。
• 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 — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction normalized remainder，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction floor quotient multiple，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction exact quotient，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 current base square，不以小數近似替代。
• 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 — 以精確有理數重算 Fermat original signed base normalized remainder，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Fermat original signed base floor quotient multiple，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Fermat original signed base exact quotient，不以小數近似替代。
• S003 · 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。
• S002 · 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。
• 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

(27, -25, 否, 是, 否, 13, coprime_form_not_applicable_without_both_hypotheses)

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。