第 19891 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":53,"a":-17}。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":53,"a":-17}。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=53;complete trial range2..floor_sqrt=7

checked rows[divisor,quotient,remainder]=[[2, 26, 1], [3, 17, 2], [4, 13, 1], [5, 10, 3], [6, 8, 5], [7, 7, 4]];first exact factor=null;prime=是

original signed base=-17;positive modulus=53;use nonnegative gcd inputs=[17, 53]

ALL Euclidean rows[left,right,q,r]=[[17, 53, 0, 17], [53, 17, 3, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-17, 53)

Fermat original signed base exact quotient = -17/53

Fermat original signed base floor quotient multiple := mul(-1, 53)

Fermat original signed base floor quotient multiple = -53

Fermat original signed base normalized remainder := sub(-17, -53)

Fermat original signed base normalized remainder = 36

Fermat original signed base:signed integer value=-17;positive modulus=53;exact quotient=-17/53;floor q=-1

value=q*modulus+remainder;normalized remainder=36 in[0,53)

Fermat original modular exponent:original normalized base=36;original exponent=52;modulus=53

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(36, 36)

Fermat original modular exponent bit0 current base square = 1296

Fermat original modular exponent bit0 squared base reduction exact quotient := div(1296, 53)

Fermat original modular exponent bit0 squared base reduction exact quotient = 1296/53

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(24, 53)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 1272

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(1296, 1272)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 24

Fermat original modular exponent bit0 squared base reduction:signed integer value=1296;positive modulus=53;exact quotient=1296/53;floor q=24

value=q*modulus+remainder;normalized remainder=24 in[0,53)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 36, 52];low bit=0

next[accumulator,base,exponent]=[1, 24, 26];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 current base square := mul(24, 24)

Fermat original modular exponent bit1 current base square = 576

Fermat original modular exponent bit1 squared base reduction exact quotient := div(576, 53)

Fermat original modular exponent bit1 squared base reduction exact quotient = 576/53

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(10, 53)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 530

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(576, 530)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 46

Fermat original modular exponent bit1 squared base reduction:signed integer value=576;positive modulus=53;exact quotient=576/53;floor q=10

value=q*modulus+remainder;normalized remainder=46 in[0,53)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 24, 26];low bit=0

next[accumulator,base,exponent]=[1, 46, 13];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(1, 46)

Fermat original modular exponent bit2 accumulator product = 46

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(46, 53)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 46/53

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(0, 53)

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

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(46, 0)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 46

Fermat original modular exponent bit2 accumulator reduction:signed integer value=46;positive modulus=53;exact quotient=46/53;floor q=0

value=q*modulus+remainder;normalized remainder=46 in[0,53)

Fermat original modular exponent bit2 current base square := mul(46, 46)

Fermat original modular exponent bit2 current base square = 2116

Fermat original modular exponent bit2 squared base reduction exact quotient := div(2116, 53)

Fermat original modular exponent bit2 squared base reduction exact quotient = 2116/53

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(39, 53)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 2067

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(2116, 2067)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 49

Fermat original modular exponent bit2 squared base reduction:signed integer value=2116;positive modulus=53;exact quotient=2116/53;floor q=39

value=q*modulus+remainder;normalized remainder=49 in[0,53)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 46, 13];low bit=1

next[accumulator,base,exponent]=[46, 49, 6];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(49, 49)

Fermat original modular exponent bit3 current base square = 2401

Fermat original modular exponent bit3 squared base reduction exact quotient := div(2401, 53)

Fermat original modular exponent bit3 squared base reduction exact quotient = 2401/53

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(45, 53)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 2385

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(2401, 2385)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 16

Fermat original modular exponent bit3 squared base reduction:signed integer value=2401;positive modulus=53;exact quotient=2401/53;floor q=45

value=q*modulus+remainder;normalized remainder=16 in[0,53)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[46, 49, 6];low bit=0

next[accumulator,base,exponent]=[46, 16, 3];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(46, 16)

Fermat original modular exponent bit4 accumulator product = 736

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(736, 53)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 736/53

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(13, 53)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 689

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(736, 689)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 47

Fermat original modular exponent bit4 accumulator reduction:signed integer value=736;positive modulus=53;exact quotient=736/53;floor q=13

value=q*modulus+remainder;normalized remainder=47 in[0,53)

Fermat original modular exponent bit4 current base square := mul(16, 16)

Fermat original modular exponent bit4 current base square = 256

Fermat original modular exponent bit4 squared base reduction exact quotient := div(256, 53)

Fermat original modular exponent bit4 squared base reduction exact quotient = 256/53

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(4, 53)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 212

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(256, 212)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 44

Fermat original modular exponent bit4 squared base reduction:signed integer value=256;positive modulus=53;exact quotient=256/53;floor q=4

value=q*modulus+remainder;normalized remainder=44 in[0,53)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[46, 16, 3];low bit=1

next[accumulator,base,exponent]=[47, 44, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 accumulator product := mul(47, 44)

Fermat original modular exponent bit5 accumulator product = 2068

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(2068, 53)

Fermat original modular exponent bit5 accumulator reduction exact quotient = 2068/53

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(39, 53)

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 2067

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(2068, 2067)

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit5 accumulator reduction:signed integer value=2068;positive modulus=53;exact quotient=2068/53;floor q=39

value=q*modulus+remainder;normalized remainder=1 in[0,53)

Fermat original modular exponent bit5 current base square := mul(44, 44)

Fermat original modular exponent bit5 current base square = 1936

Fermat original modular exponent bit5 squared base reduction exact quotient := div(1936, 53)

Fermat original modular exponent bit5 squared base reduction exact quotient = 1936/53

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(36, 53)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 1908

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(1936, 1908)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 28

Fermat original modular exponent bit5 squared base reduction:signed integer value=1936;positive modulus=53;exact quotient=1936/53;floor q=36

value=q*modulus+remainder;normalized remainder=28 in[0,53)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[47, 44, 1];low bit=1

next[accumulator,base,exponent]=[1, 28, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^52 modulo53=1

Fermat actual unit inverse product := mul(36, 28)

Fermat actual unit inverse product = 1008

Fermat actual inverse identity exact quotient := div(1008, 53)

Fermat actual inverse identity exact quotient = 1008/53

Fermat actual inverse identity floor quotient multiple := mul(19, 53)

Fermat actual inverse identity floor quotient multiple = 1007

Fermat actual inverse identity normalized remainder := sub(1008, 1007)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=1008;positive modulus=53;exact quotient=1008/53;floor q=19

value=q*modulus+remainder;normalized remainder=1 in[0,53)

VERIFIED prime p=53;nonzero original base residue=36;ACTUAL inverse witness=28

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=53;ALL nonzero residues permuted by multiplication by a=36

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

• 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 — 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 — 以精確有理數重算 Fermat actual inverse identity normalized remainder，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 floor quotient multiple，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 exact quotient，不以小數近似替代。
• S061 · 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。
• 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 — 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 — 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。
• S056 · 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。
• S055 · 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。
• S054 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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 — 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• 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

(53, 36, 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。