第 19884 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":23,"a":10}。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":23,"a":10}。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=23;complete trial range2..floor_sqrt=4

checked rows[divisor,quotient,remainder]=[[2, 11, 1], [3, 7, 2], [4, 5, 3]];first exact factor=null;prime=是

original signed base=10;positive modulus=23;use nonnegative gcd inputs=[10, 23]

ALL Euclidean rows[left,right,q,r]=[[10, 23, 0, 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(10, 23)

Fermat original signed base exact quotient = 10/23

Fermat original signed base floor quotient multiple := mul(0, 23)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(10, 0)

Fermat original signed base normalized remainder = 10

Fermat original signed base:signed integer value=10;positive modulus=23;exact quotient=10/23;floor q=0

value=q*modulus+remainder;normalized remainder=10 in[0,23)

Fermat original modular exponent:original normalized base=10;original exponent=22;modulus=23

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(10, 10)

Fermat original modular exponent bit0 current base square = 100

Fermat original modular exponent bit0 squared base reduction exact quotient := div(100, 23)

Fermat original modular exponent bit0 squared base reduction exact quotient = 100/23

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(4, 23)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 92

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(100, 92)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 8

Fermat original modular exponent bit0 squared base reduction:signed integer value=100;positive modulus=23;exact quotient=100/23;floor q=4

value=q*modulus+remainder;normalized remainder=8 in[0,23)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 10, 22];low bit=0

next[accumulator,base,exponent]=[1, 8, 11];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 8)

Fermat original modular exponent bit1 accumulator product = 8

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(8, 23)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 8/23

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 23)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(8, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 8

Fermat original modular exponent bit1 accumulator reduction:signed integer value=8;positive modulus=23;exact quotient=8/23;floor q=0

value=q*modulus+remainder;normalized remainder=8 in[0,23)

Fermat original modular exponent bit1 current base square := mul(8, 8)

Fermat original modular exponent bit1 current base square = 64

Fermat original modular exponent bit1 squared base reduction exact quotient := div(64, 23)

Fermat original modular exponent bit1 squared base reduction exact quotient = 64/23

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(2, 23)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 46

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(64, 46)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 18

Fermat original modular exponent bit1 squared base reduction:signed integer value=64;positive modulus=23;exact quotient=64/23;floor q=2

value=q*modulus+remainder;normalized remainder=18 in[0,23)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 8, 11];low bit=1

next[accumulator,base,exponent]=[8, 18, 5];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(8, 18)

Fermat original modular exponent bit2 accumulator product = 144

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(144, 23)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 144/23

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(6, 23)

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple = 138

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(144, 138)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 6

Fermat original modular exponent bit2 accumulator reduction:signed integer value=144;positive modulus=23;exact quotient=144/23;floor q=6

value=q*modulus+remainder;normalized remainder=6 in[0,23)

Fermat original modular exponent bit2 current base square := mul(18, 18)

Fermat original modular exponent bit2 current base square = 324

Fermat original modular exponent bit2 squared base reduction exact quotient := div(324, 23)

Fermat original modular exponent bit2 squared base reduction exact quotient = 324/23

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(14, 23)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 322

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(324, 322)

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

Fermat original modular exponent bit2 squared base reduction:signed integer value=324;positive modulus=23;exact quotient=324/23;floor q=14

value=q*modulus+remainder;normalized remainder=2 in[0,23)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[8, 18, 5];low bit=1

next[accumulator,base,exponent]=[6, 2, 2];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, 23)

Fermat original modular exponent bit3 squared base reduction exact quotient = 4/23

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(0, 23)

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=23;exact quotient=4/23;floor q=0

value=q*modulus+remainder;normalized remainder=4 in[0,23)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[6, 2, 2];low bit=0

next[accumulator,base,exponent]=[6, 4, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(6, 4)

Fermat original modular exponent bit4 accumulator product = 24

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(24, 23)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 24/23

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(1, 23)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 23

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(24, 23)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit4 accumulator reduction:signed integer value=24;positive modulus=23;exact quotient=24/23;floor q=1

value=q*modulus+remainder;normalized remainder=1 in[0,23)

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, 23)

Fermat original modular exponent bit4 squared base reduction exact quotient = 16/23

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(0, 23)

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=23;exact quotient=16/23;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,23)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[6, 4, 1];low bit=1

next[accumulator,base,exponent]=[1, 16, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^22 modulo23=1

Fermat actual unit inverse product := mul(10, 7)

Fermat actual unit inverse product = 70

Fermat actual inverse identity exact quotient := div(70, 23)

Fermat actual inverse identity exact quotient = 70/23

Fermat actual inverse identity floor quotient multiple := mul(3, 23)

Fermat actual inverse identity floor quotient multiple = 69

Fermat actual inverse identity normalized remainder := sub(70, 69)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=70;positive modulus=23;exact quotient=70/23;floor q=3

value=q*modulus+remainder;normalized remainder=1 in[0,23)

VERIFIED prime p=23;nonzero original base residue=10;ACTUAL inverse witness=7

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=23;ALL nonzero residues permuted by multiplication by a=10

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

• S061 · 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。
• 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 — 以精確有理數重算 Fermat actual inverse identity normalized remainder，不以小數近似替代。
• S057 · 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。
• S056 · 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。
• S055 · 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。
• 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 — 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 — 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• 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

(23, 10, 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。