第 19883 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":19,"a":-9}。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":19,"a":-9}。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=19;complete trial range2..floor_sqrt=4

checked rows[divisor,quotient,remainder]=[[2, 9, 1], [3, 6, 1], [4, 4, 3]];first exact factor=null;prime=是

original signed base=-9;positive modulus=19;use nonnegative gcd inputs=[9, 19]

ALL Euclidean rows[left,right,q,r]=[[9, 19, 0, 9], [19, 9, 2, 1], [9, 1, 9, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(-9, 19)

Fermat original signed base exact quotient = -9/19

Fermat original signed base floor quotient multiple := mul(-1, 19)

Fermat original signed base floor quotient multiple = -19

Fermat original signed base normalized remainder := sub(-9, -19)

Fermat original signed base normalized remainder = 10

Fermat original signed base:signed integer value=-9;positive modulus=19;exact quotient=-9/19;floor q=-1

value=q*modulus+remainder;normalized remainder=10 in[0,19)

Fermat original modular exponent:original normalized base=10;original exponent=18;modulus=19

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, 19)

Fermat original modular exponent bit0 squared base reduction exact quotient = 100/19

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(5, 19)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 95

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(100, 95)

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

Fermat original modular exponent bit0 squared base reduction:signed integer value=100;positive modulus=19;exact quotient=100/19;floor q=5

value=q*modulus+remainder;normalized remainder=5 in[0,19)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 10, 18];low bit=0

next[accumulator,base,exponent]=[1, 5, 9];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(1, 5)

Fermat original modular exponent bit1 accumulator product = 5

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(5, 19)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 5/19

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(0, 19)

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

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(5, 0)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 5

Fermat original modular exponent bit1 accumulator reduction:signed integer value=5;positive modulus=19;exact quotient=5/19;floor q=0

value=q*modulus+remainder;normalized remainder=5 in[0,19)

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, 19)

Fermat original modular exponent bit1 squared base reduction exact quotient = 25/19

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(1, 19)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 19

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(25, 19)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 6

Fermat original modular exponent bit1 squared base reduction:signed integer value=25;positive modulus=19;exact quotient=25/19;floor q=1

value=q*modulus+remainder;normalized remainder=6 in[0,19)

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

next[accumulator,base,exponent]=[5, 6, 4];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(6, 6)

Fermat original modular exponent bit2 current base square = 36

Fermat original modular exponent bit2 squared base reduction exact quotient := div(36, 19)

Fermat original modular exponent bit2 squared base reduction exact quotient = 36/19

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(1, 19)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 19

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(36, 19)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 17

Fermat original modular exponent bit2 squared base reduction:signed integer value=36;positive modulus=19;exact quotient=36/19;floor q=1

value=q*modulus+remainder;normalized remainder=17 in[0,19)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[5, 6, 4];low bit=0

next[accumulator,base,exponent]=[5, 17, 2];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(17, 17)

Fermat original modular exponent bit3 current base square = 289

Fermat original modular exponent bit3 squared base reduction exact quotient := div(289, 19)

Fermat original modular exponent bit3 squared base reduction exact quotient = 289/19

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(15, 19)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 285

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(289, 285)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 4

Fermat original modular exponent bit3 squared base reduction:signed integer value=289;positive modulus=19;exact quotient=289/19;floor q=15

value=q*modulus+remainder;normalized remainder=4 in[0,19)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[5, 17, 2];low bit=0

next[accumulator,base,exponent]=[5, 4, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(5, 4)

Fermat original modular exponent bit4 accumulator product = 20

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(20, 19)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 20/19

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(1, 19)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 19

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(20, 19)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit4 accumulator reduction:signed integer value=20;positive modulus=19;exact quotient=20/19;floor q=1

value=q*modulus+remainder;normalized remainder=1 in[0,19)

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, 19)

Fermat original modular exponent bit4 squared base reduction exact quotient = 16/19

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(0, 19)

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=19;exact quotient=16/19;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,19)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[5, 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^18 modulo19=1

Fermat actual unit inverse product := mul(10, 2)

Fermat actual unit inverse product = 20

Fermat actual inverse identity exact quotient := div(20, 19)

Fermat actual inverse identity exact quotient = 20/19

Fermat actual inverse identity floor quotient multiple := mul(1, 19)

Fermat actual inverse identity floor quotient multiple = 19

Fermat actual inverse identity normalized remainder := sub(20, 19)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=20;positive modulus=19;exact quotient=20/19;floor q=1

value=q*modulus+remainder;normalized remainder=1 in[0,19)

VERIFIED prime p=19;nonzero original base residue=10;ACTUAL inverse witness=2

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=19;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

• S056 · 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。
• S055 · 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。
• 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 — 以精確有理數重算 Fermat actual inverse identity normalized remainder，不以小數近似替代。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• 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

(19, 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。