第 19885 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":29,"a":-11}。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":29,"a":-11}。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=29;complete trial range2..floor_sqrt=5

checked rows[divisor,quotient,remainder]=[[2, 14, 1], [3, 9, 2], [4, 7, 1], [5, 5, 4]];first exact factor=null;prime=是

original signed base=-11;positive modulus=29;use nonnegative gcd inputs=[11, 29]

ALL Euclidean rows[left,right,q,r]=[[11, 29, 0, 11], [29, 11, 2, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-11, 29)

Fermat original signed base exact quotient = -11/29

Fermat original signed base floor quotient multiple := mul(-1, 29)

Fermat original signed base floor quotient multiple = -29

Fermat original signed base normalized remainder := sub(-11, -29)

Fermat original signed base normalized remainder = 18

Fermat original signed base:signed integer value=-11;positive modulus=29;exact quotient=-11/29;floor q=-1

value=q*modulus+remainder;normalized remainder=18 in[0,29)

Fermat original modular exponent:original normalized base=18;original exponent=28;modulus=29

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(18, 18)

Fermat original modular exponent bit0 current base square = 324

Fermat original modular exponent bit0 squared base reduction exact quotient := div(324, 29)

Fermat original modular exponent bit0 squared base reduction exact quotient = 324/29

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(11, 29)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 319

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(324, 319)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 5

Fermat original modular exponent bit0 squared base reduction:signed integer value=324;positive modulus=29;exact quotient=324/29;floor q=11

value=q*modulus+remainder;normalized remainder=5 in[0,29)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 18, 28];low bit=0

next[accumulator,base,exponent]=[1, 5, 14];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 current base square := mul(5, 5)

Fermat original modular exponent bit1 current base square = 25

Fermat original modular exponent bit1 squared base reduction exact quotient := div(25, 29)

Fermat original modular exponent bit1 squared base reduction exact quotient = 25/29

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 29)

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

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(25, 0)

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

Fermat original modular exponent bit1 squared base reduction:signed integer value=25;positive modulus=29;exact quotient=25/29;floor q=0

value=q*modulus+remainder;normalized remainder=25 in[0,29)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 5, 14];low bit=0

next[accumulator,base,exponent]=[1, 25, 7];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(1, 25)

Fermat original modular exponent bit2 accumulator product = 25

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(25, 29)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 25/29

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(0, 29)

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

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(25, 0)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 25

Fermat original modular exponent bit2 accumulator reduction:signed integer value=25;positive modulus=29;exact quotient=25/29;floor q=0

value=q*modulus+remainder;normalized remainder=25 in[0,29)

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, 29)

Fermat original modular exponent bit2 squared base reduction exact quotient = 625/29

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(21, 29)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 609

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(625, 609)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 16

Fermat original modular exponent bit2 squared base reduction:signed integer value=625;positive modulus=29;exact quotient=625/29;floor q=21

value=q*modulus+remainder;normalized remainder=16 in[0,29)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 25, 7];low bit=1

next[accumulator,base,exponent]=[25, 16, 3];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 accumulator product := mul(25, 16)

Fermat original modular exponent bit3 accumulator product = 400

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(400, 29)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 400/29

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(13, 29)

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple = 377

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(400, 377)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 23

Fermat original modular exponent bit3 accumulator reduction:signed integer value=400;positive modulus=29;exact quotient=400/29;floor q=13

value=q*modulus+remainder;normalized remainder=23 in[0,29)

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, 29)

Fermat original modular exponent bit3 squared base reduction exact quotient = 256/29

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(8, 29)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 232

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(256, 232)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 24

Fermat original modular exponent bit3 squared base reduction:signed integer value=256;positive modulus=29;exact quotient=256/29;floor q=8

value=q*modulus+remainder;normalized remainder=24 in[0,29)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[25, 16, 3];low bit=1

next[accumulator,base,exponent]=[23, 24, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(23, 24)

Fermat original modular exponent bit4 accumulator product = 552

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(552, 29)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 552/29

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(19, 29)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 551

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(552, 551)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit4 accumulator reduction:signed integer value=552;positive modulus=29;exact quotient=552/29;floor q=19

value=q*modulus+remainder;normalized remainder=1 in[0,29)

Fermat original modular exponent bit4 current base square := mul(24, 24)

Fermat original modular exponent bit4 current base square = 576

Fermat original modular exponent bit4 squared base reduction exact quotient := div(576, 29)

Fermat original modular exponent bit4 squared base reduction exact quotient = 576/29

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(19, 29)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 551

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(576, 551)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 25

Fermat original modular exponent bit4 squared base reduction:signed integer value=576;positive modulus=29;exact quotient=576/29;floor q=19

value=q*modulus+remainder;normalized remainder=25 in[0,29)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[23, 24, 1];low bit=1

next[accumulator,base,exponent]=[1, 25, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^28 modulo29=1

Fermat actual unit inverse product := mul(18, 21)

Fermat actual unit inverse product = 378

Fermat actual inverse identity exact quotient := div(378, 29)

Fermat actual inverse identity exact quotient = 378/29

Fermat actual inverse identity floor quotient multiple := mul(13, 29)

Fermat actual inverse identity floor quotient multiple = 377

Fermat actual inverse identity normalized remainder := sub(378, 377)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=378;positive modulus=29;exact quotient=378/29;floor q=13

value=q*modulus+remainder;normalized remainder=1 in[0,29)

VERIFIED prime p=29;nonzero original base residue=18;ACTUAL inverse witness=21

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=29;ALL nonzero residues permuted by multiplication by a=18

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 — 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• 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

(29, 18, 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。