第 19887 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":37,"a":-13}。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":37,"a":-13}。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=37;complete trial range2..floor_sqrt=6

checked rows[divisor,quotient,remainder]=[[2, 18, 1], [3, 12, 1], [4, 9, 1], [5, 7, 2], [6, 6, 1]];first exact factor=null;prime=是

original signed base=-13;positive modulus=37;use nonnegative gcd inputs=[13, 37]

ALL Euclidean rows[left,right,q,r]=[[13, 37, 0, 13], [37, 13, 2, 11], [13, 11, 1, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-13, 37)

Fermat original signed base exact quotient = -13/37

Fermat original signed base floor quotient multiple := mul(-1, 37)

Fermat original signed base floor quotient multiple = -37

Fermat original signed base normalized remainder := sub(-13, -37)

Fermat original signed base normalized remainder = 24

Fermat original signed base:signed integer value=-13;positive modulus=37;exact quotient=-13/37;floor q=-1

value=q*modulus+remainder;normalized remainder=24 in[0,37)

Fermat original modular exponent:original normalized base=24;original exponent=36;modulus=37

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(24, 24)

Fermat original modular exponent bit0 current base square = 576

Fermat original modular exponent bit0 squared base reduction exact quotient := div(576, 37)

Fermat original modular exponent bit0 squared base reduction exact quotient = 576/37

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(15, 37)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 555

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(576, 555)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 21

Fermat original modular exponent bit0 squared base reduction:signed integer value=576;positive modulus=37;exact quotient=576/37;floor q=15

value=q*modulus+remainder;normalized remainder=21 in[0,37)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 24, 36];low bit=0

next[accumulator,base,exponent]=[1, 21, 18];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 current base square := mul(21, 21)

Fermat original modular exponent bit1 current base square = 441

Fermat original modular exponent bit1 squared base reduction exact quotient := div(441, 37)

Fermat original modular exponent bit1 squared base reduction exact quotient = 441/37

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(11, 37)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 407

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(441, 407)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 34

Fermat original modular exponent bit1 squared base reduction:signed integer value=441;positive modulus=37;exact quotient=441/37;floor q=11

value=q*modulus+remainder;normalized remainder=34 in[0,37)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 21, 18];low bit=0

next[accumulator,base,exponent]=[1, 34, 9];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(1, 34)

Fermat original modular exponent bit2 accumulator product = 34

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(34, 37)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 34/37

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(0, 37)

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(34, 0)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 34

Fermat original modular exponent bit2 accumulator reduction:signed integer value=34;positive modulus=37;exact quotient=34/37;floor q=0

value=q*modulus+remainder;normalized remainder=34 in[0,37)

Fermat original modular exponent bit2 current base square := mul(34, 34)

Fermat original modular exponent bit2 current base square = 1156

Fermat original modular exponent bit2 squared base reduction exact quotient := div(1156, 37)

Fermat original modular exponent bit2 squared base reduction exact quotient = 1156/37

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(31, 37)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 1147

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(1156, 1147)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 9

Fermat original modular exponent bit2 squared base reduction:signed integer value=1156;positive modulus=37;exact quotient=1156/37;floor q=31

value=q*modulus+remainder;normalized remainder=9 in[0,37)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 34, 9];low bit=1

next[accumulator,base,exponent]=[34, 9, 4];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(9, 9)

Fermat original modular exponent bit3 current base square = 81

Fermat original modular exponent bit3 squared base reduction exact quotient := div(81, 37)

Fermat original modular exponent bit3 squared base reduction exact quotient = 81/37

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(2, 37)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 74

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(81, 74)

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

Fermat original modular exponent bit3 squared base reduction:signed integer value=81;positive modulus=37;exact quotient=81/37;floor q=2

value=q*modulus+remainder;normalized remainder=7 in[0,37)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[34, 9, 4];low bit=0

next[accumulator,base,exponent]=[34, 7, 2];split exponent=2*q+bit preserves original-power invariant

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, 37)

Fermat original modular exponent bit4 squared base reduction exact quotient = 49/37

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(1, 37)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 37

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(49, 37)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 12

Fermat original modular exponent bit4 squared base reduction:signed integer value=49;positive modulus=37;exact quotient=49/37;floor q=1

value=q*modulus+remainder;normalized remainder=12 in[0,37)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[34, 7, 2];low bit=0

next[accumulator,base,exponent]=[34, 12, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 accumulator product := mul(34, 12)

Fermat original modular exponent bit5 accumulator product = 408

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(408, 37)

Fermat original modular exponent bit5 accumulator reduction exact quotient = 408/37

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(11, 37)

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 407

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(408, 407)

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit5 accumulator reduction:signed integer value=408;positive modulus=37;exact quotient=408/37;floor q=11

value=q*modulus+remainder;normalized remainder=1 in[0,37)

Fermat original modular exponent bit5 current base square := mul(12, 12)

Fermat original modular exponent bit5 current base square = 144

Fermat original modular exponent bit5 squared base reduction exact quotient := div(144, 37)

Fermat original modular exponent bit5 squared base reduction exact quotient = 144/37

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(3, 37)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 111

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(144, 111)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 33

Fermat original modular exponent bit5 squared base reduction:signed integer value=144;positive modulus=37;exact quotient=144/37;floor q=3

value=q*modulus+remainder;normalized remainder=33 in[0,37)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[34, 12, 1];low bit=1

next[accumulator,base,exponent]=[1, 33, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^36 modulo37=1

Fermat actual unit inverse product := mul(24, 17)

Fermat actual unit inverse product = 408

Fermat actual inverse identity exact quotient := div(408, 37)

Fermat actual inverse identity exact quotient = 408/37

Fermat actual inverse identity floor quotient multiple := mul(11, 37)

Fermat actual inverse identity floor quotient multiple = 407

Fermat actual inverse identity normalized remainder := sub(408, 407)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=408;positive modulus=37;exact quotient=408/37;floor q=11

value=q*modulus+remainder;normalized remainder=1 in[0,37)

VERIFIED prime p=37;nonzero original base residue=24;ACTUAL inverse witness=17

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=37;ALL nonzero residues permuted by multiplication by a=24

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 — 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 — 以精確有理數重算 Fermat original modular exponent bit2 accumulator 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 accumulator 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 accumulator 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 accumulator product，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

(37, 24, 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。