第 19899 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":89,"a":-25}。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":89,"a":-25}。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=89;complete trial range2..floor_sqrt=9

checked rows[divisor,quotient,remainder]=[[2, 44, 1], [3, 29, 2], [4, 22, 1], [5, 17, 4], [6, 14, 5], [7, 12, 5], [8, 11, 1], [9, 9, 8]];first exact factor=null;prime=是

original signed base=-25;positive modulus=89;use nonnegative gcd inputs=[25, 89]

ALL Euclidean rows[left,right,q,r]=[[25, 89, 0, 25], [89, 25, 3, 14], [25, 14, 1, 11], [14, 11, 1, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-25, 89)

Fermat original signed base exact quotient = -25/89

Fermat original signed base floor quotient multiple := mul(-1, 89)

Fermat original signed base floor quotient multiple = -89

Fermat original signed base normalized remainder := sub(-25, -89)

Fermat original signed base normalized remainder = 64

Fermat original signed base:signed integer value=-25;positive modulus=89;exact quotient=-25/89;floor q=-1

value=q*modulus+remainder;normalized remainder=64 in[0,89)

Fermat original modular exponent:original normalized base=64;original exponent=88;modulus=89

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(64, 64)

Fermat original modular exponent bit0 current base square = 4096

Fermat original modular exponent bit0 squared base reduction exact quotient := div(4096, 89)

Fermat original modular exponent bit0 squared base reduction exact quotient = 4096/89

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(46, 89)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 4094

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(4096, 4094)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 2

Fermat original modular exponent bit0 squared base reduction:signed integer value=4096;positive modulus=89;exact quotient=4096/89;floor q=46

value=q*modulus+remainder;normalized remainder=2 in[0,89)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 64, 88];low bit=0

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

Fermat original modular exponent bit1 current base square := mul(2, 2)

Fermat original modular exponent bit1 current base square = 4

Fermat original modular exponent bit1 squared base reduction exact quotient := div(4, 89)

Fermat original modular exponent bit1 squared base reduction exact quotient = 4/89

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 89)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(4, 0)

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

Fermat original modular exponent bit1 squared base reduction:signed integer value=4;positive modulus=89;exact quotient=4/89;floor q=0

value=q*modulus+remainder;normalized remainder=4 in[0,89)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 2, 44];low bit=0

next[accumulator,base,exponent]=[1, 4, 22];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, 89)

Fermat original modular exponent bit2 squared base reduction exact quotient = 16/89

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(0, 89)

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=89;exact quotient=16/89;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,89)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 4, 22];low bit=0

next[accumulator,base,exponent]=[1, 16, 11];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 accumulator product := mul(1, 16)

Fermat original modular exponent bit3 accumulator product = 16

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(16, 89)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 16/89

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(0, 89)

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

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(16, 0)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 16

Fermat original modular exponent bit3 accumulator reduction:signed integer value=16;positive modulus=89;exact quotient=16/89;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,89)

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, 89)

Fermat original modular exponent bit3 squared base reduction exact quotient = 256/89

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(2, 89)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 178

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(256, 178)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 78

Fermat original modular exponent bit3 squared base reduction:signed integer value=256;positive modulus=89;exact quotient=256/89;floor q=2

value=q*modulus+remainder;normalized remainder=78 in[0,89)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 16, 11];low bit=1

next[accumulator,base,exponent]=[16, 78, 5];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(16, 78)

Fermat original modular exponent bit4 accumulator product = 1248

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(1248, 89)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 1248/89

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(14, 89)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 1246

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(1248, 1246)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 2

Fermat original modular exponent bit4 accumulator reduction:signed integer value=1248;positive modulus=89;exact quotient=1248/89;floor q=14

value=q*modulus+remainder;normalized remainder=2 in[0,89)

Fermat original modular exponent bit4 current base square := mul(78, 78)

Fermat original modular exponent bit4 current base square = 6084

Fermat original modular exponent bit4 squared base reduction exact quotient := div(6084, 89)

Fermat original modular exponent bit4 squared base reduction exact quotient = 6084/89

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(68, 89)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 6052

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(6084, 6052)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 32

Fermat original modular exponent bit4 squared base reduction:signed integer value=6084;positive modulus=89;exact quotient=6084/89;floor q=68

value=q*modulus+remainder;normalized remainder=32 in[0,89)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[16, 78, 5];low bit=1

next[accumulator,base,exponent]=[2, 32, 2];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 current base square := mul(32, 32)

Fermat original modular exponent bit5 current base square = 1024

Fermat original modular exponent bit5 squared base reduction exact quotient := div(1024, 89)

Fermat original modular exponent bit5 squared base reduction exact quotient = 1024/89

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(11, 89)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 979

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(1024, 979)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 45

Fermat original modular exponent bit5 squared base reduction:signed integer value=1024;positive modulus=89;exact quotient=1024/89;floor q=11

value=q*modulus+remainder;normalized remainder=45 in[0,89)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[2, 32, 2];low bit=0

next[accumulator,base,exponent]=[2, 45, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit6 accumulator product := mul(2, 45)

Fermat original modular exponent bit6 accumulator product = 90

Fermat original modular exponent bit6 accumulator reduction exact quotient := div(90, 89)

Fermat original modular exponent bit6 accumulator reduction exact quotient = 90/89

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple := mul(1, 89)

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple = 89

Fermat original modular exponent bit6 accumulator reduction normalized remainder := sub(90, 89)

Fermat original modular exponent bit6 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit6 accumulator reduction:signed integer value=90;positive modulus=89;exact quotient=90/89;floor q=1

value=q*modulus+remainder;normalized remainder=1 in[0,89)

Fermat original modular exponent bit6 current base square := mul(45, 45)

Fermat original modular exponent bit6 current base square = 2025

Fermat original modular exponent bit6 squared base reduction exact quotient := div(2025, 89)

Fermat original modular exponent bit6 squared base reduction exact quotient = 2025/89

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(22, 89)

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 1958

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(2025, 1958)

Fermat original modular exponent bit6 squared base reduction normalized remainder = 67

Fermat original modular exponent bit6 squared base reduction:signed integer value=2025;positive modulus=89;exact quotient=2025/89;floor q=22

value=q*modulus+remainder;normalized remainder=67 in[0,89)

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[2, 45, 1];low bit=1

next[accumulator,base,exponent]=[1, 67, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^88 modulo89=1

Fermat actual unit inverse product := mul(64, 32)

Fermat actual unit inverse product = 2048

Fermat actual inverse identity exact quotient := div(2048, 89)

Fermat actual inverse identity exact quotient = 2048/89

Fermat actual inverse identity floor quotient multiple := mul(23, 89)

Fermat actual inverse identity floor quotient multiple = 2047

Fermat actual inverse identity normalized remainder := sub(2048, 2047)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=2048;positive modulus=89;exact quotient=2048/89;floor q=23

value=q*modulus+remainder;normalized remainder=1 in[0,89)

VERIFIED prime p=89;nonzero original base residue=64;ACTUAL inverse witness=32

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=89;ALL nonzero residues permuted by multiplication by a=64

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 — 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 — 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• 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 squared base 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 squared base 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 squared base 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 current base square，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

(89, 64, 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。