第 19898 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":83,"a":24}。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":83,"a":24}。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=83;complete trial range2..floor_sqrt=9

checked rows[divisor,quotient,remainder]=[[2, 41, 1], [3, 27, 2], [4, 20, 3], [5, 16, 3], [6, 13, 5], [7, 11, 6], [8, 10, 3], [9, 9, 2]];first exact factor=null;prime=是

original signed base=24;positive modulus=83;use nonnegative gcd inputs=[24, 83]

ALL Euclidean rows[left,right,q,r]=[[24, 83, 0, 24], [83, 24, 3, 11], [24, 11, 2, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(24, 83)

Fermat original signed base exact quotient = 24/83

Fermat original signed base floor quotient multiple := mul(0, 83)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(24, 0)

Fermat original signed base normalized remainder = 24

Fermat original signed base:signed integer value=24;positive modulus=83;exact quotient=24/83;floor q=0

value=q*modulus+remainder;normalized remainder=24 in[0,83)

Fermat original modular exponent:original normalized base=24;original exponent=82;modulus=83

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(24, 24)

Fermat original modular exponent bit0 current base square = 576

Fermat original modular exponent bit0 squared base reduction exact quotient := div(576, 83)

Fermat original modular exponent bit0 squared base reduction exact quotient = 576/83

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(6, 83)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 498

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(576, 498)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 78

Fermat original modular exponent bit0 squared base reduction:signed integer value=576;positive modulus=83;exact quotient=576/83;floor q=6

value=q*modulus+remainder;normalized remainder=78 in[0,83)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 24, 82];low bit=0

next[accumulator,base,exponent]=[1, 78, 41];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 78)

Fermat original modular exponent bit1 accumulator product = 78

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(78, 83)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 78/83

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 83)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(78, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 78

Fermat original modular exponent bit1 accumulator reduction:signed integer value=78;positive modulus=83;exact quotient=78/83;floor q=0

value=q*modulus+remainder;normalized remainder=78 in[0,83)

Fermat original modular exponent bit1 current base square := mul(78, 78)

Fermat original modular exponent bit1 current base square = 6084

Fermat original modular exponent bit1 squared base reduction exact quotient := div(6084, 83)

Fermat original modular exponent bit1 squared base reduction exact quotient = 6084/83

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(73, 83)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 6059

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(6084, 6059)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 25

Fermat original modular exponent bit1 squared base reduction:signed integer value=6084;positive modulus=83;exact quotient=6084/83;floor q=73

value=q*modulus+remainder;normalized remainder=25 in[0,83)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 78, 41];low bit=1

next[accumulator,base,exponent]=[78, 25, 20];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(25, 25)

Fermat original modular exponent bit2 current base square = 625

Fermat original modular exponent bit2 squared base reduction exact quotient := div(625, 83)

Fermat original modular exponent bit2 squared base reduction exact quotient = 625/83

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(7, 83)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 581

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(625, 581)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 44

Fermat original modular exponent bit2 squared base reduction:signed integer value=625;positive modulus=83;exact quotient=625/83;floor q=7

value=q*modulus+remainder;normalized remainder=44 in[0,83)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[78, 25, 20];low bit=0

next[accumulator,base,exponent]=[78, 44, 10];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(44, 44)

Fermat original modular exponent bit3 current base square = 1936

Fermat original modular exponent bit3 squared base reduction exact quotient := div(1936, 83)

Fermat original modular exponent bit3 squared base reduction exact quotient = 1936/83

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(23, 83)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 1909

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(1936, 1909)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 27

Fermat original modular exponent bit3 squared base reduction:signed integer value=1936;positive modulus=83;exact quotient=1936/83;floor q=23

value=q*modulus+remainder;normalized remainder=27 in[0,83)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[78, 44, 10];low bit=0

next[accumulator,base,exponent]=[78, 27, 5];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(78, 27)

Fermat original modular exponent bit4 accumulator product = 2106

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(2106, 83)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 2106/83

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(25, 83)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 2075

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(2106, 2075)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 31

Fermat original modular exponent bit4 accumulator reduction:signed integer value=2106;positive modulus=83;exact quotient=2106/83;floor q=25

value=q*modulus+remainder;normalized remainder=31 in[0,83)

Fermat original modular exponent bit4 current base square := mul(27, 27)

Fermat original modular exponent bit4 current base square = 729

Fermat original modular exponent bit4 squared base reduction exact quotient := div(729, 83)

Fermat original modular exponent bit4 squared base reduction exact quotient = 729/83

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(8, 83)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 664

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(729, 664)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 65

Fermat original modular exponent bit4 squared base reduction:signed integer value=729;positive modulus=83;exact quotient=729/83;floor q=8

value=q*modulus+remainder;normalized remainder=65 in[0,83)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[78, 27, 5];low bit=1

next[accumulator,base,exponent]=[31, 65, 2];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 current base square := mul(65, 65)

Fermat original modular exponent bit5 current base square = 4225

Fermat original modular exponent bit5 squared base reduction exact quotient := div(4225, 83)

Fermat original modular exponent bit5 squared base reduction exact quotient = 4225/83

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(50, 83)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 4150

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(4225, 4150)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 75

Fermat original modular exponent bit5 squared base reduction:signed integer value=4225;positive modulus=83;exact quotient=4225/83;floor q=50

value=q*modulus+remainder;normalized remainder=75 in[0,83)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[31, 65, 2];low bit=0

next[accumulator,base,exponent]=[31, 75, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit6 accumulator product := mul(31, 75)

Fermat original modular exponent bit6 accumulator product = 2325

Fermat original modular exponent bit6 accumulator reduction exact quotient := div(2325, 83)

Fermat original modular exponent bit6 accumulator reduction exact quotient = 2325/83

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple := mul(28, 83)

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple = 2324

Fermat original modular exponent bit6 accumulator reduction normalized remainder := sub(2325, 2324)

Fermat original modular exponent bit6 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit6 accumulator reduction:signed integer value=2325;positive modulus=83;exact quotient=2325/83;floor q=28

value=q*modulus+remainder;normalized remainder=1 in[0,83)

Fermat original modular exponent bit6 current base square := mul(75, 75)

Fermat original modular exponent bit6 current base square = 5625

Fermat original modular exponent bit6 squared base reduction exact quotient := div(5625, 83)

Fermat original modular exponent bit6 squared base reduction exact quotient = 5625/83

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(67, 83)

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 5561

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(5625, 5561)

Fermat original modular exponent bit6 squared base reduction normalized remainder = 64

Fermat original modular exponent bit6 squared base reduction:signed integer value=5625;positive modulus=83;exact quotient=5625/83;floor q=67

value=q*modulus+remainder;normalized remainder=64 in[0,83)

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[31, 75, 1];low bit=1

next[accumulator,base,exponent]=[1, 64, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^82 modulo83=1

Fermat actual unit inverse product := mul(24, 45)

Fermat actual unit inverse product = 1080

Fermat actual inverse identity exact quotient := div(1080, 83)

Fermat actual inverse identity exact quotient = 1080/83

Fermat actual inverse identity floor quotient multiple := mul(13, 83)

Fermat actual inverse identity floor quotient multiple = 1079

Fermat actual inverse identity normalized remainder := sub(1080, 1079)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=1080;positive modulus=83;exact quotient=1080/83;floor q=13

value=q*modulus+remainder;normalized remainder=1 in[0,83)

VERIFIED prime p=83;nonzero original base residue=24;ACTUAL inverse witness=45

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=83;ALL nonzero residues permuted by multiplication by a=24

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 — 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

(83, 24, 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。