第 19886 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":31,"a":12}。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":31,"a":12}。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=31;complete trial range2..floor_sqrt=5

checked rows[divisor,quotient,remainder]=[[2, 15, 1], [3, 10, 1], [4, 7, 3], [5, 6, 1]];first exact factor=null;prime=是

original signed base=12;positive modulus=31;use nonnegative gcd inputs=[12, 31]

ALL Euclidean rows[left,right,q,r]=[[12, 31, 0, 12], [31, 12, 2, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(12, 31)

Fermat original signed base exact quotient = 12/31

Fermat original signed base floor quotient multiple := mul(0, 31)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(12, 0)

Fermat original signed base normalized remainder = 12

Fermat original signed base:signed integer value=12;positive modulus=31;exact quotient=12/31;floor q=0

value=q*modulus+remainder;normalized remainder=12 in[0,31)

Fermat original modular exponent:original normalized base=12;original exponent=30;modulus=31

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(12, 12)

Fermat original modular exponent bit0 current base square = 144

Fermat original modular exponent bit0 squared base reduction exact quotient := div(144, 31)

Fermat original modular exponent bit0 squared base reduction exact quotient = 144/31

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(4, 31)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 124

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(144, 124)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 20

Fermat original modular exponent bit0 squared base reduction:signed integer value=144;positive modulus=31;exact quotient=144/31;floor q=4

value=q*modulus+remainder;normalized remainder=20 in[0,31)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 12, 30];low bit=0

next[accumulator,base,exponent]=[1, 20, 15];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 20)

Fermat original modular exponent bit1 accumulator product = 20

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(20, 31)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 20/31

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 31)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(20, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 20

Fermat original modular exponent bit1 accumulator reduction:signed integer value=20;positive modulus=31;exact quotient=20/31;floor q=0

value=q*modulus+remainder;normalized remainder=20 in[0,31)

Fermat original modular exponent bit1 current base square := mul(20, 20)

Fermat original modular exponent bit1 current base square = 400

Fermat original modular exponent bit1 squared base reduction exact quotient := div(400, 31)

Fermat original modular exponent bit1 squared base reduction exact quotient = 400/31

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(12, 31)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 372

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(400, 372)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 28

Fermat original modular exponent bit1 squared base reduction:signed integer value=400;positive modulus=31;exact quotient=400/31;floor q=12

value=q*modulus+remainder;normalized remainder=28 in[0,31)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 20, 15];low bit=1

next[accumulator,base,exponent]=[20, 28, 7];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(20, 28)

Fermat original modular exponent bit2 accumulator product = 560

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(560, 31)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 560/31

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(18, 31)

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple = 558

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(560, 558)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 2

Fermat original modular exponent bit2 accumulator reduction:signed integer value=560;positive modulus=31;exact quotient=560/31;floor q=18

value=q*modulus+remainder;normalized remainder=2 in[0,31)

Fermat original modular exponent bit2 current base square := mul(28, 28)

Fermat original modular exponent bit2 current base square = 784

Fermat original modular exponent bit2 squared base reduction exact quotient := div(784, 31)

Fermat original modular exponent bit2 squared base reduction exact quotient = 784/31

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(25, 31)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 775

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(784, 775)

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

Fermat original modular exponent bit2 squared base reduction:signed integer value=784;positive modulus=31;exact quotient=784/31;floor q=25

value=q*modulus+remainder;normalized remainder=9 in[0,31)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[20, 28, 7];low bit=1

next[accumulator,base,exponent]=[2, 9, 3];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 accumulator product := mul(2, 9)

Fermat original modular exponent bit3 accumulator product = 18

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(18, 31)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 18/31

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(0, 31)

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

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(18, 0)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 18

Fermat original modular exponent bit3 accumulator reduction:signed integer value=18;positive modulus=31;exact quotient=18/31;floor q=0

value=q*modulus+remainder;normalized remainder=18 in[0,31)

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, 31)

Fermat original modular exponent bit3 squared base reduction exact quotient = 81/31

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(2, 31)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 62

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(81, 62)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 19

Fermat original modular exponent bit3 squared base reduction:signed integer value=81;positive modulus=31;exact quotient=81/31;floor q=2

value=q*modulus+remainder;normalized remainder=19 in[0,31)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[2, 9, 3];low bit=1

next[accumulator,base,exponent]=[18, 19, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(18, 19)

Fermat original modular exponent bit4 accumulator product = 342

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(342, 31)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 342/31

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(11, 31)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 341

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(342, 341)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit4 accumulator reduction:signed integer value=342;positive modulus=31;exact quotient=342/31;floor q=11

value=q*modulus+remainder;normalized remainder=1 in[0,31)

Fermat original modular exponent bit4 current base square := mul(19, 19)

Fermat original modular exponent bit4 current base square = 361

Fermat original modular exponent bit4 squared base reduction exact quotient := div(361, 31)

Fermat original modular exponent bit4 squared base reduction exact quotient = 361/31

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(11, 31)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 341

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(361, 341)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 20

Fermat original modular exponent bit4 squared base reduction:signed integer value=361;positive modulus=31;exact quotient=361/31;floor q=11

value=q*modulus+remainder;normalized remainder=20 in[0,31)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[18, 19, 1];low bit=1

next[accumulator,base,exponent]=[1, 20, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^30 modulo31=1

Fermat actual unit inverse product := mul(12, 13)

Fermat actual unit inverse product = 156

Fermat actual inverse identity exact quotient := div(156, 31)

Fermat actual inverse identity exact quotient = 156/31

Fermat actual inverse identity floor quotient multiple := mul(5, 31)

Fermat actual inverse identity floor quotient multiple = 155

Fermat actual inverse identity normalized remainder := sub(156, 155)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=156;positive modulus=31;exact quotient=156/31;floor q=5

value=q*modulus+remainder;normalized remainder=1 in[0,31)

VERIFIED prime p=31;nonzero original base residue=12;ACTUAL inverse witness=13

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=31;ALL nonzero residues permuted by multiplication by a=12

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

• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• S054 · 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。
• S053 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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 — 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 Fermat original modular exponent bit3 squared base reduction normalized remainder，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 bit3 squared base reduction floor quotient multiple，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 bit3 squared base reduction exact quotient，不以小數近似替代。
• S042 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• S031 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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

(31, 12, 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。