第 19888 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-prime-coprime；資料={"p":41,"a":14}。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-prime-coprime；資料={"p":41,"a":14}。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-prime-coprime;original snapshot and mathematics retained

explicit ordered source output fields=[p, normalized_base, power_p_minus_1, congruence_certificate, boundary]

ACTUAL integer modulus=41;complete trial range2..floor_sqrt=6

checked rows[divisor,quotient,remainder]=[[2, 20, 1], [3, 13, 2], [4, 10, 1], [5, 8, 1], [6, 6, 5]];first exact factor=null;prime=是

original signed base=14;positive modulus=41;use nonnegative gcd inputs=[14, 41]

ALL Euclidean rows[left,right,q,r]=[[14, 41, 0, 14], [41, 14, 2, 13], [14, 13, 1, 1], [13, 1, 13, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(14, 41)

Fermat original signed base exact quotient = 14/41

Fermat original signed base floor quotient multiple := mul(0, 41)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(14, 0)

Fermat original signed base normalized remainder = 14

Fermat original signed base:signed integer value=14;positive modulus=41;exact quotient=14/41;floor q=0

value=q*modulus+remainder;normalized remainder=14 in[0,41)

Fermat original modular exponent:original normalized base=14;original exponent=40;modulus=41

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(14, 14)

Fermat original modular exponent bit0 current base square = 196

Fermat original modular exponent bit0 squared base reduction exact quotient := div(196, 41)

Fermat original modular exponent bit0 squared base reduction exact quotient = 196/41

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(4, 41)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 164

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(196, 164)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 32

Fermat original modular exponent bit0 squared base reduction:signed integer value=196;positive modulus=41;exact quotient=196/41;floor q=4

value=q*modulus+remainder;normalized remainder=32 in[0,41)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 14, 40];low bit=0

next[accumulator,base,exponent]=[1, 32, 20];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 current base square := mul(32, 32)

Fermat original modular exponent bit1 current base square = 1024

Fermat original modular exponent bit1 squared base reduction exact quotient := div(1024, 41)

Fermat original modular exponent bit1 squared base reduction exact quotient = 1024/41

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(24, 41)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 984

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(1024, 984)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 40

Fermat original modular exponent bit1 squared base reduction:signed integer value=1024;positive modulus=41;exact quotient=1024/41;floor q=24

value=q*modulus+remainder;normalized remainder=40 in[0,41)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 32, 20];low bit=0

next[accumulator,base,exponent]=[1, 40, 10];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(40, 40)

Fermat original modular exponent bit2 current base square = 1600

Fermat original modular exponent bit2 squared base reduction exact quotient := div(1600, 41)

Fermat original modular exponent bit2 squared base reduction exact quotient = 1600/41

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(39, 41)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 1599

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(1600, 1599)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 1

Fermat original modular exponent bit2 squared base reduction:signed integer value=1600;positive modulus=41;exact quotient=1600/41;floor q=39

value=q*modulus+remainder;normalized remainder=1 in[0,41)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 40, 10];low bit=0

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

Fermat original modular exponent bit3 accumulator product := mul(1, 1)

Fermat original modular exponent bit3 accumulator product = 1

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(1, 41)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 1/41

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(0, 41)

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit3 accumulator reduction:signed integer value=1;positive modulus=41;exact quotient=1/41;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,41)

Fermat original modular exponent bit3 current base square := mul(1, 1)

Fermat original modular exponent bit3 current base square = 1

Fermat original modular exponent bit3 squared base reduction exact quotient := div(1, 41)

Fermat original modular exponent bit3 squared base reduction exact quotient = 1/41

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(0, 41)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 1

Fermat original modular exponent bit3 squared base reduction:signed integer value=1;positive modulus=41;exact quotient=1/41;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,41)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 1, 5];low bit=1

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

Fermat original modular exponent bit4 current base square := mul(1, 1)

Fermat original modular exponent bit4 current base square = 1

Fermat original modular exponent bit4 squared base reduction exact quotient := div(1, 41)

Fermat original modular exponent bit4 squared base reduction exact quotient = 1/41

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(0, 41)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 1

Fermat original modular exponent bit4 squared base reduction:signed integer value=1;positive modulus=41;exact quotient=1/41;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,41)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[1, 1, 2];low bit=0

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

Fermat original modular exponent bit5 accumulator product := mul(1, 1)

Fermat original modular exponent bit5 accumulator product = 1

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(1, 41)

Fermat original modular exponent bit5 accumulator reduction exact quotient = 1/41

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(0, 41)

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit5 accumulator reduction:signed integer value=1;positive modulus=41;exact quotient=1/41;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,41)

Fermat original modular exponent bit5 current base square := mul(1, 1)

Fermat original modular exponent bit5 current base square = 1

Fermat original modular exponent bit5 squared base reduction exact quotient := div(1, 41)

Fermat original modular exponent bit5 squared base reduction exact quotient = 1/41

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(0, 41)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 1

Fermat original modular exponent bit5 squared base reduction:signed integer value=1;positive modulus=41;exact quotient=1/41;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,41)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[1, 1, 1];low bit=1

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

Fermat original modular exponent:remaining exponent0;final normalized accumulator=1

EXACT original base^40 modulo41=1

Fermat actual unit inverse product := mul(14, 3)

Fermat actual unit inverse product = 42

Fermat actual inverse identity exact quotient := div(42, 41)

Fermat actual inverse identity exact quotient = 42/41

Fermat actual inverse identity floor quotient multiple := mul(1, 41)

Fermat actual inverse identity floor quotient multiple = 41

Fermat actual inverse identity normalized remainder := sub(42, 41)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=42;positive modulus=41;exact quotient=42/41;floor q=1

value=q*modulus+remainder;normalized remainder=1 in[0,41)

VERIFIED prime p=41;nonzero original base residue=14;ACTUAL inverse witness=3

multiplication r->a*r mod p permutes ALL nonzero residues1..p-1:inverse gives injectivity and finite self-injection gives surjectivity

same verified prime p=41;ALL nonzero residues permuted by multiplication by a=14

product over r=1..p-1 gives a^(p-1)*(p-1)! congruent to (p-1)!;EVERY factor is a prime-field unit,so factorial is a UNIT;cancel ONLY this nonzero unit to obtain a^(p-1) congruent1

逐步反向核對

• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · premise_and_result_audit — 以精確有理數重算 Fermat actual inverse identity normalized remainder，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Fermat actual inverse identity floor quotient multiple，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Fermat actual inverse identity exact quotient，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Fermat actual unit inverse product，不以小數近似替代。
• 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 Fermat original modular exponent bit5 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 Fermat original modular exponent bit5 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 Fermat original modular exponent bit5 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 Fermat original modular exponent bit5 current 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 Fermat original modular exponent bit5 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 Fermat original modular exponent bit5 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 Fermat original modular exponent bit5 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 Fermat original modular exponent bit5 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 Fermat original modular exponent bit4 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 Fermat original modular exponent bit4 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 Fermat original modular exponent bit4 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 Fermat original modular exponent bit4 current 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 Fermat original modular exponent bit3 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 Fermat original modular exponent bit3 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 Fermat original modular exponent bit3 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 Fermat original modular exponent bit3 current 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 Fermat original modular exponent bit3 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 Fermat original modular exponent bit3 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 Fermat original modular exponent bit3 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 Fermat original modular exponent bit3 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 Fermat original modular exponent bit2 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 Fermat original modular exponent bit2 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 Fermat original modular exponent bit2 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 Fermat original modular exponent bit2 current 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 Fermat original modular exponent bit1 squared base 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 squared base 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 squared base 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 current base square，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

(41, 14, 1, 是, prime_coprime_form)

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。