第 19897 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":79,"a":-23}。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":79,"a":-23}。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=79;complete trial range2..floor_sqrt=8

checked rows[divisor,quotient,remainder]=[[2, 39, 1], [3, 26, 1], [4, 19, 3], [5, 15, 4], [6, 13, 1], [7, 11, 2], [8, 9, 7]];first exact factor=null;prime=是

original signed base=-23;positive modulus=79;use nonnegative gcd inputs=[23, 79]

ALL Euclidean rows[left,right,q,r]=[[23, 79, 0, 23], [79, 23, 3, 10], [23, 10, 2, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-23, 79)

Fermat original signed base exact quotient = -23/79

Fermat original signed base floor quotient multiple := mul(-1, 79)

Fermat original signed base floor quotient multiple = -79

Fermat original signed base normalized remainder := sub(-23, -79)

Fermat original signed base normalized remainder = 56

Fermat original signed base:signed integer value=-23;positive modulus=79;exact quotient=-23/79;floor q=-1

value=q*modulus+remainder;normalized remainder=56 in[0,79)

Fermat original modular exponent:original normalized base=56;original exponent=78;modulus=79

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(56, 56)

Fermat original modular exponent bit0 current base square = 3136

Fermat original modular exponent bit0 squared base reduction exact quotient := div(3136, 79)

Fermat original modular exponent bit0 squared base reduction exact quotient = 3136/79

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(39, 79)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 3081

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(3136, 3081)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 55

Fermat original modular exponent bit0 squared base reduction:signed integer value=3136;positive modulus=79;exact quotient=3136/79;floor q=39

value=q*modulus+remainder;normalized remainder=55 in[0,79)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 56, 78];low bit=0

next[accumulator,base,exponent]=[1, 55, 39];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 55)

Fermat original modular exponent bit1 accumulator product = 55

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(55, 79)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 55/79

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 79)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(55, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 55

Fermat original modular exponent bit1 accumulator reduction:signed integer value=55;positive modulus=79;exact quotient=55/79;floor q=0

value=q*modulus+remainder;normalized remainder=55 in[0,79)

Fermat original modular exponent bit1 current base square := mul(55, 55)

Fermat original modular exponent bit1 current base square = 3025

Fermat original modular exponent bit1 squared base reduction exact quotient := div(3025, 79)

Fermat original modular exponent bit1 squared base reduction exact quotient = 3025/79

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(38, 79)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 3002

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(3025, 3002)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 23

Fermat original modular exponent bit1 squared base reduction:signed integer value=3025;positive modulus=79;exact quotient=3025/79;floor q=38

value=q*modulus+remainder;normalized remainder=23 in[0,79)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 55, 39];low bit=1

next[accumulator,base,exponent]=[55, 23, 19];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(55, 23)

Fermat original modular exponent bit2 accumulator product = 1265

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(1265, 79)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 1265/79

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(16, 79)

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple = 1264

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(1265, 1264)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit2 accumulator reduction:signed integer value=1265;positive modulus=79;exact quotient=1265/79;floor q=16

value=q*modulus+remainder;normalized remainder=1 in[0,79)

Fermat original modular exponent bit2 current base square := mul(23, 23)

Fermat original modular exponent bit2 current base square = 529

Fermat original modular exponent bit2 squared base reduction exact quotient := div(529, 79)

Fermat original modular exponent bit2 squared base reduction exact quotient = 529/79

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(6, 79)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 474

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(529, 474)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 55

Fermat original modular exponent bit2 squared base reduction:signed integer value=529;positive modulus=79;exact quotient=529/79;floor q=6

value=q*modulus+remainder;normalized remainder=55 in[0,79)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[55, 23, 19];low bit=1

next[accumulator,base,exponent]=[1, 55, 9];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 accumulator product := mul(1, 55)

Fermat original modular exponent bit3 accumulator product = 55

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(55, 79)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 55/79

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(0, 79)

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

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(55, 0)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 55

Fermat original modular exponent bit3 accumulator reduction:signed integer value=55;positive modulus=79;exact quotient=55/79;floor q=0

value=q*modulus+remainder;normalized remainder=55 in[0,79)

Fermat original modular exponent bit3 current base square := mul(55, 55)

Fermat original modular exponent bit3 current base square = 3025

Fermat original modular exponent bit3 squared base reduction exact quotient := div(3025, 79)

Fermat original modular exponent bit3 squared base reduction exact quotient = 3025/79

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(38, 79)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 3002

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(3025, 3002)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 23

Fermat original modular exponent bit3 squared base reduction:signed integer value=3025;positive modulus=79;exact quotient=3025/79;floor q=38

value=q*modulus+remainder;normalized remainder=23 in[0,79)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 55, 9];low bit=1

next[accumulator,base,exponent]=[55, 23, 4];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 current base square := mul(23, 23)

Fermat original modular exponent bit4 current base square = 529

Fermat original modular exponent bit4 squared base reduction exact quotient := div(529, 79)

Fermat original modular exponent bit4 squared base reduction exact quotient = 529/79

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(6, 79)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 474

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(529, 474)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 55

Fermat original modular exponent bit4 squared base reduction:signed integer value=529;positive modulus=79;exact quotient=529/79;floor q=6

value=q*modulus+remainder;normalized remainder=55 in[0,79)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[55, 23, 4];low bit=0

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

Fermat original modular exponent bit5 current base square := mul(55, 55)

Fermat original modular exponent bit5 current base square = 3025

Fermat original modular exponent bit5 squared base reduction exact quotient := div(3025, 79)

Fermat original modular exponent bit5 squared base reduction exact quotient = 3025/79

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(38, 79)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 3002

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(3025, 3002)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 23

Fermat original modular exponent bit5 squared base reduction:signed integer value=3025;positive modulus=79;exact quotient=3025/79;floor q=38

value=q*modulus+remainder;normalized remainder=23 in[0,79)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[55, 55, 2];low bit=0

next[accumulator,base,exponent]=[55, 23, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit6 accumulator product := mul(55, 23)

Fermat original modular exponent bit6 accumulator product = 1265

Fermat original modular exponent bit6 accumulator reduction exact quotient := div(1265, 79)

Fermat original modular exponent bit6 accumulator reduction exact quotient = 1265/79

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple := mul(16, 79)

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple = 1264

Fermat original modular exponent bit6 accumulator reduction normalized remainder := sub(1265, 1264)

Fermat original modular exponent bit6 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit6 accumulator reduction:signed integer value=1265;positive modulus=79;exact quotient=1265/79;floor q=16

value=q*modulus+remainder;normalized remainder=1 in[0,79)

Fermat original modular exponent bit6 current base square := mul(23, 23)

Fermat original modular exponent bit6 current base square = 529

Fermat original modular exponent bit6 squared base reduction exact quotient := div(529, 79)

Fermat original modular exponent bit6 squared base reduction exact quotient = 529/79

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(6, 79)

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 474

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(529, 474)

Fermat original modular exponent bit6 squared base reduction normalized remainder = 55

Fermat original modular exponent bit6 squared base reduction:signed integer value=529;positive modulus=79;exact quotient=529/79;floor q=6

value=q*modulus+remainder;normalized remainder=55 in[0,79)

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[55, 23, 1];low bit=1

next[accumulator,base,exponent]=[1, 55, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^78 modulo79=1

Fermat actual unit inverse product := mul(56, 24)

Fermat actual unit inverse product = 1344

Fermat actual inverse identity exact quotient := div(1344, 79)

Fermat actual inverse identity exact quotient = 1344/79

Fermat actual inverse identity floor quotient multiple := mul(17, 79)

Fermat actual inverse identity floor quotient multiple = 1343

Fermat actual inverse identity normalized remainder := sub(1344, 1343)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=1344;positive modulus=79;exact quotient=1344/79;floor q=17

value=q*modulus+remainder;normalized remainder=1 in[0,79)

VERIFIED prime p=79;nonzero original base residue=56;ACTUAL inverse witness=24

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=79;ALL nonzero residues permuted by multiplication by a=56

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

• S078 · 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。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator 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 — 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 — 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 squared base 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 squared base 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 squared base 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 current base square，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

(79, 56, 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。