第 19892 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":59,"a":18}。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":59,"a":18}。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=59;complete trial range2..floor_sqrt=7

checked rows[divisor,quotient,remainder]=[[2, 29, 1], [3, 19, 2], [4, 14, 3], [5, 11, 4], [6, 9, 5], [7, 8, 3]];first exact factor=null;prime=是

original signed base=18;positive modulus=59;use nonnegative gcd inputs=[18, 59]

ALL Euclidean rows[left,right,q,r]=[[18, 59, 0, 18], [59, 18, 3, 5], [18, 5, 3, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(18, 59)

Fermat original signed base exact quotient = 18/59

Fermat original signed base floor quotient multiple := mul(0, 59)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(18, 0)

Fermat original signed base normalized remainder = 18

Fermat original signed base:signed integer value=18;positive modulus=59;exact quotient=18/59;floor q=0

value=q*modulus+remainder;normalized remainder=18 in[0,59)

Fermat original modular exponent:original normalized base=18;original exponent=58;modulus=59

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, 59)

Fermat original modular exponent bit0 squared base reduction exact quotient = 324/59

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(5, 59)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 295

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(324, 295)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 29

Fermat original modular exponent bit0 squared base reduction:signed integer value=324;positive modulus=59;exact quotient=324/59;floor q=5

value=q*modulus+remainder;normalized remainder=29 in[0,59)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 18, 58];low bit=0

next[accumulator,base,exponent]=[1, 29, 29];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 29)

Fermat original modular exponent bit1 accumulator product = 29

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(29, 59)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 29/59

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 59)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(29, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 29

Fermat original modular exponent bit1 accumulator reduction:signed integer value=29;positive modulus=59;exact quotient=29/59;floor q=0

value=q*modulus+remainder;normalized remainder=29 in[0,59)

Fermat original modular exponent bit1 current base square := mul(29, 29)

Fermat original modular exponent bit1 current base square = 841

Fermat original modular exponent bit1 squared base reduction exact quotient := div(841, 59)

Fermat original modular exponent bit1 squared base reduction exact quotient = 841/59

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(14, 59)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 826

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(841, 826)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 15

Fermat original modular exponent bit1 squared base reduction:signed integer value=841;positive modulus=59;exact quotient=841/59;floor q=14

value=q*modulus+remainder;normalized remainder=15 in[0,59)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 29, 29];low bit=1

next[accumulator,base,exponent]=[29, 15, 14];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(15, 15)

Fermat original modular exponent bit2 current base square = 225

Fermat original modular exponent bit2 squared base reduction exact quotient := div(225, 59)

Fermat original modular exponent bit2 squared base reduction exact quotient = 225/59

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(3, 59)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 177

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(225, 177)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 48

Fermat original modular exponent bit2 squared base reduction:signed integer value=225;positive modulus=59;exact quotient=225/59;floor q=3

value=q*modulus+remainder;normalized remainder=48 in[0,59)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[29, 15, 14];low bit=0

next[accumulator,base,exponent]=[29, 48, 7];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 accumulator product := mul(29, 48)

Fermat original modular exponent bit3 accumulator product = 1392

Fermat original modular exponent bit3 accumulator reduction exact quotient := div(1392, 59)

Fermat original modular exponent bit3 accumulator reduction exact quotient = 1392/59

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple := mul(23, 59)

Fermat original modular exponent bit3 accumulator reduction floor quotient multiple = 1357

Fermat original modular exponent bit3 accumulator reduction normalized remainder := sub(1392, 1357)

Fermat original modular exponent bit3 accumulator reduction normalized remainder = 35

Fermat original modular exponent bit3 accumulator reduction:signed integer value=1392;positive modulus=59;exact quotient=1392/59;floor q=23

value=q*modulus+remainder;normalized remainder=35 in[0,59)

Fermat original modular exponent bit3 current base square := mul(48, 48)

Fermat original modular exponent bit3 current base square = 2304

Fermat original modular exponent bit3 squared base reduction exact quotient := div(2304, 59)

Fermat original modular exponent bit3 squared base reduction exact quotient = 2304/59

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(39, 59)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 2301

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(2304, 2301)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 3

Fermat original modular exponent bit3 squared base reduction:signed integer value=2304;positive modulus=59;exact quotient=2304/59;floor q=39

value=q*modulus+remainder;normalized remainder=3 in[0,59)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[29, 48, 7];low bit=1

next[accumulator,base,exponent]=[35, 3, 3];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(35, 3)

Fermat original modular exponent bit4 accumulator product = 105

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(105, 59)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 105/59

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(1, 59)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 59

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(105, 59)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 46

Fermat original modular exponent bit4 accumulator reduction:signed integer value=105;positive modulus=59;exact quotient=105/59;floor q=1

value=q*modulus+remainder;normalized remainder=46 in[0,59)

Fermat original modular exponent bit4 current base square := mul(3, 3)

Fermat original modular exponent bit4 current base square = 9

Fermat original modular exponent bit4 squared base reduction exact quotient := div(9, 59)

Fermat original modular exponent bit4 squared base reduction exact quotient = 9/59

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(0, 59)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(9, 0)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 9

Fermat original modular exponent bit4 squared base reduction:signed integer value=9;positive modulus=59;exact quotient=9/59;floor q=0

value=q*modulus+remainder;normalized remainder=9 in[0,59)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[35, 3, 3];low bit=1

next[accumulator,base,exponent]=[46, 9, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 accumulator product := mul(46, 9)

Fermat original modular exponent bit5 accumulator product = 414

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(414, 59)

Fermat original modular exponent bit5 accumulator reduction exact quotient = 414/59

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(7, 59)

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 413

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(414, 413)

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit5 accumulator reduction:signed integer value=414;positive modulus=59;exact quotient=414/59;floor q=7

value=q*modulus+remainder;normalized remainder=1 in[0,59)

Fermat original modular exponent bit5 current base square := mul(9, 9)

Fermat original modular exponent bit5 current base square = 81

Fermat original modular exponent bit5 squared base reduction exact quotient := div(81, 59)

Fermat original modular exponent bit5 squared base reduction exact quotient = 81/59

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(1, 59)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 59

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(81, 59)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 22

Fermat original modular exponent bit5 squared base reduction:signed integer value=81;positive modulus=59;exact quotient=81/59;floor q=1

value=q*modulus+remainder;normalized remainder=22 in[0,59)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[46, 9, 1];low bit=1

next[accumulator,base,exponent]=[1, 22, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^58 modulo59=1

Fermat actual unit inverse product := mul(18, 23)

Fermat actual unit inverse product = 414

Fermat actual inverse identity exact quotient := div(414, 59)

Fermat actual inverse identity exact quotient = 414/59

Fermat actual inverse identity floor quotient multiple := mul(7, 59)

Fermat actual inverse identity floor quotient multiple = 413

Fermat actual inverse identity normalized remainder := sub(414, 413)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=414;positive modulus=59;exact quotient=414/59;floor q=7

value=q*modulus+remainder;normalized remainder=1 in[0,59)

VERIFIED prime p=59;nonzero original base residue=18;ACTUAL inverse witness=23

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=59;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

• 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 — 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 — 以精確有理數重算 Fermat actual inverse identity normalized remainder，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 floor quotient multiple，不以小數近似替代。
• S067 · 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。
• S066 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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 — 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 — 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

(59, 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。