第 19895 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":71,"a":-21}。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":71,"a":-21}。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=71;complete trial range2..floor_sqrt=8

checked rows[divisor,quotient,remainder]=[[2, 35, 1], [3, 23, 2], [4, 17, 3], [5, 14, 1], [6, 11, 5], [7, 10, 1], [8, 8, 7]];first exact factor=null;prime=是

original signed base=-21;positive modulus=71;use nonnegative gcd inputs=[21, 71]

ALL Euclidean rows[left,right,q,r]=[[21, 71, 0, 21], [71, 21, 3, 8], [21, 8, 2, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-21, 71)

Fermat original signed base exact quotient = -21/71

Fermat original signed base floor quotient multiple := mul(-1, 71)

Fermat original signed base floor quotient multiple = -71

Fermat original signed base normalized remainder := sub(-21, -71)

Fermat original signed base normalized remainder = 50

Fermat original signed base:signed integer value=-21;positive modulus=71;exact quotient=-21/71;floor q=-1

value=q*modulus+remainder;normalized remainder=50 in[0,71)

Fermat original modular exponent:original normalized base=50;original exponent=70;modulus=71

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(50, 50)

Fermat original modular exponent bit0 current base square = 2500

Fermat original modular exponent bit0 squared base reduction exact quotient := div(2500, 71)

Fermat original modular exponent bit0 squared base reduction exact quotient = 2500/71

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(35, 71)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 2485

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(2500, 2485)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 15

Fermat original modular exponent bit0 squared base reduction:signed integer value=2500;positive modulus=71;exact quotient=2500/71;floor q=35

value=q*modulus+remainder;normalized remainder=15 in[0,71)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 50, 70];low bit=0

next[accumulator,base,exponent]=[1, 15, 35];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 15)

Fermat original modular exponent bit1 accumulator product = 15

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(15, 71)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 15/71

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 71)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(15, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 15

Fermat original modular exponent bit1 accumulator reduction:signed integer value=15;positive modulus=71;exact quotient=15/71;floor q=0

value=q*modulus+remainder;normalized remainder=15 in[0,71)

Fermat original modular exponent bit1 current base square := mul(15, 15)

Fermat original modular exponent bit1 current base square = 225

Fermat original modular exponent bit1 squared base reduction exact quotient := div(225, 71)

Fermat original modular exponent bit1 squared base reduction exact quotient = 225/71

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(3, 71)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 213

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(225, 213)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 12

Fermat original modular exponent bit1 squared base reduction:signed integer value=225;positive modulus=71;exact quotient=225/71;floor q=3

value=q*modulus+remainder;normalized remainder=12 in[0,71)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 15, 35];low bit=1

next[accumulator,base,exponent]=[15, 12, 17];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(15, 12)

Fermat original modular exponent bit2 accumulator product = 180

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(180, 71)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 180/71

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(2, 71)

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple = 142

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(180, 142)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 38

Fermat original modular exponent bit2 accumulator reduction:signed integer value=180;positive modulus=71;exact quotient=180/71;floor q=2

value=q*modulus+remainder;normalized remainder=38 in[0,71)

Fermat original modular exponent bit2 current base square := mul(12, 12)

Fermat original modular exponent bit2 current base square = 144

Fermat original modular exponent bit2 squared base reduction exact quotient := div(144, 71)

Fermat original modular exponent bit2 squared base reduction exact quotient = 144/71

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(2, 71)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 142

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(144, 142)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 2

Fermat original modular exponent bit2 squared base reduction:signed integer value=144;positive modulus=71;exact quotient=144/71;floor q=2

value=q*modulus+remainder;normalized remainder=2 in[0,71)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[15, 12, 17];low bit=1

next[accumulator,base,exponent]=[38, 2, 8];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(2, 2)

Fermat original modular exponent bit3 current base square = 4

Fermat original modular exponent bit3 squared base reduction exact quotient := div(4, 71)

Fermat original modular exponent bit3 squared base reduction exact quotient = 4/71

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(0, 71)

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

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(4, 0)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 4

Fermat original modular exponent bit3 squared base reduction:signed integer value=4;positive modulus=71;exact quotient=4/71;floor q=0

value=q*modulus+remainder;normalized remainder=4 in[0,71)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[38, 2, 8];low bit=0

next[accumulator,base,exponent]=[38, 4, 4];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 current base square := mul(4, 4)

Fermat original modular exponent bit4 current base square = 16

Fermat original modular exponent bit4 squared base reduction exact quotient := div(16, 71)

Fermat original modular exponent bit4 squared base reduction exact quotient = 16/71

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(0, 71)

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

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(16, 0)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 16

Fermat original modular exponent bit4 squared base reduction:signed integer value=16;positive modulus=71;exact quotient=16/71;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,71)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[38, 4, 4];low bit=0

next[accumulator,base,exponent]=[38, 16, 2];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 current base square := mul(16, 16)

Fermat original modular exponent bit5 current base square = 256

Fermat original modular exponent bit5 squared base reduction exact quotient := div(256, 71)

Fermat original modular exponent bit5 squared base reduction exact quotient = 256/71

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(3, 71)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 213

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(256, 213)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 43

Fermat original modular exponent bit5 squared base reduction:signed integer value=256;positive modulus=71;exact quotient=256/71;floor q=3

value=q*modulus+remainder;normalized remainder=43 in[0,71)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[38, 16, 2];low bit=0

next[accumulator,base,exponent]=[38, 43, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit6 accumulator product := mul(38, 43)

Fermat original modular exponent bit6 accumulator product = 1634

Fermat original modular exponent bit6 accumulator reduction exact quotient := div(1634, 71)

Fermat original modular exponent bit6 accumulator reduction exact quotient = 1634/71

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple := mul(23, 71)

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple = 1633

Fermat original modular exponent bit6 accumulator reduction normalized remainder := sub(1634, 1633)

Fermat original modular exponent bit6 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit6 accumulator reduction:signed integer value=1634;positive modulus=71;exact quotient=1634/71;floor q=23

value=q*modulus+remainder;normalized remainder=1 in[0,71)

Fermat original modular exponent bit6 current base square := mul(43, 43)

Fermat original modular exponent bit6 current base square = 1849

Fermat original modular exponent bit6 squared base reduction exact quotient := div(1849, 71)

Fermat original modular exponent bit6 squared base reduction exact quotient = 1849/71

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(26, 71)

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 1846

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(1849, 1846)

Fermat original modular exponent bit6 squared base reduction normalized remainder = 3

Fermat original modular exponent bit6 squared base reduction:signed integer value=1849;positive modulus=71;exact quotient=1849/71;floor q=26

value=q*modulus+remainder;normalized remainder=3 in[0,71)

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[38, 43, 1];low bit=1

next[accumulator,base,exponent]=[1, 3, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^70 modulo71=1

Fermat actual unit inverse product := mul(50, 27)

Fermat actual unit inverse product = 1350

Fermat actual inverse identity exact quotient := div(1350, 71)

Fermat actual inverse identity exact quotient = 1350/71

Fermat actual inverse identity floor quotient multiple := mul(19, 71)

Fermat actual inverse identity floor quotient multiple = 1349

Fermat actual inverse identity normalized remainder := sub(1350, 1349)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=1350;positive modulus=71;exact quotient=1350/71;floor q=19

value=q*modulus+remainder;normalized remainder=1 in[0,71)

VERIFIED prime p=71;nonzero original base residue=50;ACTUAL inverse witness=27

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=71;ALL nonzero residues permuted by multiplication by a=50

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

• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• 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 original modular exponent bit6 squared base reduction normalized remainder，不以小數近似替代。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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 — 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 — 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 squared base 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 squared base 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 squared base 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 current base square，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 — 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

(71, 50, 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。