第 19878 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":5,"a":4}。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":5,"a":4}。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=5;complete trial range2..floor_sqrt=2

checked rows[divisor,quotient,remainder]=[[2, 2, 1]];first exact factor=null;prime=是

original signed base=4;positive modulus=5;use nonnegative gcd inputs=[4, 5]

ALL Euclidean rows[left,right,q,r]=[[4, 5, 0, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(4, 5)

Fermat original signed base exact quotient = 4/5

Fermat original signed base floor quotient multiple := mul(0, 5)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(4, 0)

Fermat original signed base normalized remainder = 4

Fermat original signed base:signed integer value=4;positive modulus=5;exact quotient=4/5;floor q=0

value=q*modulus+remainder;normalized remainder=4 in[0,5)

Fermat original modular exponent:original normalized base=4;original exponent=4;modulus=5

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(4, 4)

Fermat original modular exponent bit0 current base square = 16

Fermat original modular exponent bit0 squared base reduction exact quotient := div(16, 5)

Fermat original modular exponent bit0 squared base reduction exact quotient = 16/5

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(3, 5)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 15

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(16, 15)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 1

Fermat original modular exponent bit0 squared base reduction:signed integer value=16;positive modulus=5;exact quotient=16/5;floor q=3

value=q*modulus+remainder;normalized remainder=1 in[0,5)

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

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

Fermat original modular exponent bit1 current base square := mul(1, 1)

Fermat original modular exponent bit1 current base square = 1

Fermat original modular exponent bit1 squared base reduction exact quotient := div(1, 5)

Fermat original modular exponent bit1 squared base reduction exact quotient = 1/5

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 5)

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

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 1

Fermat original modular exponent bit1 squared base reduction:signed integer value=1;positive modulus=5;exact quotient=1/5;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,5)

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

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

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

Fermat original modular exponent bit2 accumulator product = 1

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(1, 5)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 1/5

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(0, 5)

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

Fermat original modular exponent bit2 accumulator reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit2 accumulator reduction:signed integer value=1;positive modulus=5;exact quotient=1/5;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,5)

Fermat original modular exponent bit2 current base square := mul(1, 1)

Fermat original modular exponent bit2 current base square = 1

Fermat original modular exponent bit2 squared base reduction exact quotient := div(1, 5)

Fermat original modular exponent bit2 squared base reduction exact quotient = 1/5

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(0, 5)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(1, 0)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 1

Fermat original modular exponent bit2 squared base reduction:signed integer value=1;positive modulus=5;exact quotient=1/5;floor q=0

value=q*modulus+remainder;normalized remainder=1 in[0,5)

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

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

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

EXACT original base^4 modulo5=1

Fermat actual unit inverse product := mul(4, 4)

Fermat actual unit inverse product = 16

Fermat actual inverse identity exact quotient := div(16, 5)

Fermat actual inverse identity exact quotient = 16/5

Fermat actual inverse identity floor quotient multiple := mul(3, 5)

Fermat actual inverse identity floor quotient multiple = 15

Fermat actual inverse identity normalized remainder := sub(16, 15)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=16;positive modulus=5;exact quotient=16/5;floor q=3

value=q*modulus+remainder;normalized remainder=1 in[0,5)

VERIFIED prime p=5;nonzero original base residue=4;ACTUAL inverse witness=4

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=5;ALL nonzero residues permuted by multiplication by a=4

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

• S039 · 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。
• S038 · 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。
• 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 — 以精確有理數重算 Fermat actual inverse identity normalized remainder，不以小數近似替代。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• 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

(5, 4, 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。