第 19894 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":67,"a":20}。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":67,"a":20}。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=67;complete trial range2..floor_sqrt=8

checked rows[divisor,quotient,remainder]=[[2, 33, 1], [3, 22, 1], [4, 16, 3], [5, 13, 2], [6, 11, 1], [7, 9, 4], [8, 8, 3]];first exact factor=null;prime=是

original signed base=20;positive modulus=67;use nonnegative gcd inputs=[20, 67]

ALL Euclidean rows[left,right,q,r]=[[20, 67, 0, 20], [67, 20, 3, 7], [20, 7, 2, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(20, 67)

Fermat original signed base exact quotient = 20/67

Fermat original signed base floor quotient multiple := mul(0, 67)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(20, 0)

Fermat original signed base normalized remainder = 20

Fermat original signed base:signed integer value=20;positive modulus=67;exact quotient=20/67;floor q=0

value=q*modulus+remainder;normalized remainder=20 in[0,67)

Fermat original modular exponent:original normalized base=20;original exponent=66;modulus=67

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(20, 20)

Fermat original modular exponent bit0 current base square = 400

Fermat original modular exponent bit0 squared base reduction exact quotient := div(400, 67)

Fermat original modular exponent bit0 squared base reduction exact quotient = 400/67

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(5, 67)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 335

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(400, 335)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 65

Fermat original modular exponent bit0 squared base reduction:signed integer value=400;positive modulus=67;exact quotient=400/67;floor q=5

value=q*modulus+remainder;normalized remainder=65 in[0,67)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 20, 66];low bit=0

next[accumulator,base,exponent]=[1, 65, 33];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 65)

Fermat original modular exponent bit1 accumulator product = 65

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(65, 67)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 65/67

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 67)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(65, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 65

Fermat original modular exponent bit1 accumulator reduction:signed integer value=65;positive modulus=67;exact quotient=65/67;floor q=0

value=q*modulus+remainder;normalized remainder=65 in[0,67)

Fermat original modular exponent bit1 current base square := mul(65, 65)

Fermat original modular exponent bit1 current base square = 4225

Fermat original modular exponent bit1 squared base reduction exact quotient := div(4225, 67)

Fermat original modular exponent bit1 squared base reduction exact quotient = 4225/67

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(63, 67)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 4221

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(4225, 4221)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 4

Fermat original modular exponent bit1 squared base reduction:signed integer value=4225;positive modulus=67;exact quotient=4225/67;floor q=63

value=q*modulus+remainder;normalized remainder=4 in[0,67)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 65, 33];low bit=1

next[accumulator,base,exponent]=[65, 4, 16];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(4, 4)

Fermat original modular exponent bit2 current base square = 16

Fermat original modular exponent bit2 squared base reduction exact quotient := div(16, 67)

Fermat original modular exponent bit2 squared base reduction exact quotient = 16/67

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(0, 67)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(16, 0)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 16

Fermat original modular exponent bit2 squared base reduction:signed integer value=16;positive modulus=67;exact quotient=16/67;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,67)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[65, 4, 16];low bit=0

next[accumulator,base,exponent]=[65, 16, 8];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(16, 16)

Fermat original modular exponent bit3 current base square = 256

Fermat original modular exponent bit3 squared base reduction exact quotient := div(256, 67)

Fermat original modular exponent bit3 squared base reduction exact quotient = 256/67

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(3, 67)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 201

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(256, 201)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 55

Fermat original modular exponent bit3 squared base reduction:signed integer value=256;positive modulus=67;exact quotient=256/67;floor q=3

value=q*modulus+remainder;normalized remainder=55 in[0,67)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[65, 16, 8];low bit=0

next[accumulator,base,exponent]=[65, 55, 4];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 current base square := mul(55, 55)

Fermat original modular exponent bit4 current base square = 3025

Fermat original modular exponent bit4 squared base reduction exact quotient := div(3025, 67)

Fermat original modular exponent bit4 squared base reduction exact quotient = 3025/67

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(45, 67)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 3015

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(3025, 3015)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 10

Fermat original modular exponent bit4 squared base reduction:signed integer value=3025;positive modulus=67;exact quotient=3025/67;floor q=45

value=q*modulus+remainder;normalized remainder=10 in[0,67)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[65, 55, 4];low bit=0

next[accumulator,base,exponent]=[65, 10, 2];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 current base square := mul(10, 10)

Fermat original modular exponent bit5 current base square = 100

Fermat original modular exponent bit5 squared base reduction exact quotient := div(100, 67)

Fermat original modular exponent bit5 squared base reduction exact quotient = 100/67

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(1, 67)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 67

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(100, 67)

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

Fermat original modular exponent bit5 squared base reduction:signed integer value=100;positive modulus=67;exact quotient=100/67;floor q=1

value=q*modulus+remainder;normalized remainder=33 in[0,67)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[65, 10, 2];low bit=0

next[accumulator,base,exponent]=[65, 33, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit6 accumulator product := mul(65, 33)

Fermat original modular exponent bit6 accumulator product = 2145

Fermat original modular exponent bit6 accumulator reduction exact quotient := div(2145, 67)

Fermat original modular exponent bit6 accumulator reduction exact quotient = 2145/67

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple := mul(32, 67)

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple = 2144

Fermat original modular exponent bit6 accumulator reduction normalized remainder := sub(2145, 2144)

Fermat original modular exponent bit6 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit6 accumulator reduction:signed integer value=2145;positive modulus=67;exact quotient=2145/67;floor q=32

value=q*modulus+remainder;normalized remainder=1 in[0,67)

Fermat original modular exponent bit6 current base square := mul(33, 33)

Fermat original modular exponent bit6 current base square = 1089

Fermat original modular exponent bit6 squared base reduction exact quotient := div(1089, 67)

Fermat original modular exponent bit6 squared base reduction exact quotient = 1089/67

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(16, 67)

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 1072

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(1089, 1072)

Fermat original modular exponent bit6 squared base reduction normalized remainder = 17

Fermat original modular exponent bit6 squared base reduction:signed integer value=1089;positive modulus=67;exact quotient=1089/67;floor q=16

value=q*modulus+remainder;normalized remainder=17 in[0,67)

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[65, 33, 1];low bit=1

next[accumulator,base,exponent]=[1, 17, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^66 modulo67=1

Fermat actual unit inverse product := mul(20, 57)

Fermat actual unit inverse product = 1140

Fermat actual inverse identity exact quotient := div(1140, 67)

Fermat actual inverse identity exact quotient = 1140/67

Fermat actual inverse identity floor quotient multiple := mul(17, 67)

Fermat actual inverse identity floor quotient multiple = 1139

Fermat actual inverse identity normalized remainder := sub(1140, 1139)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=1140;positive modulus=67;exact quotient=1140/67;floor q=17

value=q*modulus+remainder;normalized remainder=1 in[0,67)

VERIFIED prime p=67;nonzero original base residue=20;ACTUAL inverse witness=57

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=67;ALL nonzero residues permuted by multiplication by a=20

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

• S068 · 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。
• S067 · 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。
• 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 — 以精確有理數重算 Fermat actual inverse identity normalized remainder，不以小數近似替代。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• 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 — 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 — 以精確有理數重算 Fermat original modular exponent bit6 squared base reduction normalized remainder，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 squared base reduction floor quotient multiple，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 squared base reduction exact quotient，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 bit6 current base square，不以小數近似替代。
• 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 — 以精確有理數重算 Fermat original modular exponent bit6 accumulator reduction normalized remainder，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 accumulator reduction floor quotient multiple，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 bit6 accumulator reduction exact quotient，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 bit6 accumulator product，不以小數近似替代。
• S049 · 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。
• 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 squared base 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 squared base 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 squared base 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 current base square，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 — 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

(67, 20, 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。