第 19867 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-hypothesis-audit；資料={"modulus":20,"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-hypothesis-audit；資料={"modulus":20,"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-hypothesis-audit;original snapshot and mathematics retained

explicit ordered source output fields=[modulus, original_base, prime, coprime, coprime_form_applicable, power_modulus_minus_1, boundary]

ACTUAL integer modulus=20;complete trial range2..floor_sqrt=4

checked rows[divisor,quotient,remainder]=[[2, 10, 0]];first exact factor=2;prime=否

original signed base=18;positive modulus=20;use nonnegative gcd inputs=[18, 20]

ALL Euclidean rows[left,right,q,r]=[[18, 20, 0, 18], [20, 18, 1, 2], [18, 2, 9, 0]];terminal nonnegative gcd=2;coprime=否

Fermat original signed base exact quotient := div(18, 20)

Fermat original signed base exact quotient = 9/10

Fermat original signed base floor quotient multiple := mul(0, 20)

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=20;exact quotient=9/10;floor q=0

value=q*modulus+remainder;normalized remainder=18 in[0,20)

Fermat original modular exponent:original normalized base=18;original exponent=19;modulus=20

initialize accumulator1,current_base=base,remaining=exponent;invariant accumulator*current_base^remaining congruent to original power

Fermat original modular exponent bit0 accumulator product := mul(1, 18)

Fermat original modular exponent bit0 accumulator product = 18

Fermat original modular exponent bit0 accumulator reduction exact quotient := div(18, 20)

Fermat original modular exponent bit0 accumulator reduction exact quotient = 9/10

Fermat original modular exponent bit0 accumulator reduction floor quotient multiple := mul(0, 20)

Fermat original modular exponent bit0 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit0 accumulator reduction normalized remainder := sub(18, 0)

Fermat original modular exponent bit0 accumulator reduction normalized remainder = 18

Fermat original modular exponent bit0 accumulator reduction:signed integer value=18;positive modulus=20;exact quotient=9/10;floor q=0

value=q*modulus+remainder;normalized remainder=18 in[0,20)

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, 20)

Fermat original modular exponent bit0 squared base reduction exact quotient = 81/5

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(16, 20)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 320

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(324, 320)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 4

Fermat original modular exponent bit0 squared base reduction:signed integer value=324;positive modulus=20;exact quotient=81/5;floor q=16

value=q*modulus+remainder;normalized remainder=4 in[0,20)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 18, 19];low bit=1

next[accumulator,base,exponent]=[18, 4, 9];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 accumulator product := mul(18, 4)

Fermat original modular exponent bit1 accumulator product = 72

Fermat original modular exponent bit1 accumulator reduction exact quotient := div(72, 20)

Fermat original modular exponent bit1 accumulator reduction exact quotient = 18/5

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple := mul(3, 20)

Fermat original modular exponent bit1 accumulator reduction floor quotient multiple = 60

Fermat original modular exponent bit1 accumulator reduction normalized remainder := sub(72, 60)

Fermat original modular exponent bit1 accumulator reduction normalized remainder = 12

Fermat original modular exponent bit1 accumulator reduction:signed integer value=72;positive modulus=20;exact quotient=18/5;floor q=3

value=q*modulus+remainder;normalized remainder=12 in[0,20)

Fermat original modular exponent bit1 current base square := mul(4, 4)

Fermat original modular exponent bit1 current base square = 16

Fermat original modular exponent bit1 squared base reduction exact quotient := div(16, 20)

Fermat original modular exponent bit1 squared base reduction exact quotient = 4/5

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 20)

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

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(16, 0)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 16

Fermat original modular exponent bit1 squared base reduction:signed integer value=16;positive modulus=20;exact quotient=4/5;floor q=0

value=q*modulus+remainder;normalized remainder=16 in[0,20)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[18, 4, 9];low bit=1

next[accumulator,base,exponent]=[12, 16, 4];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(16, 16)

Fermat original modular exponent bit2 current base square = 256

Fermat original modular exponent bit2 squared base reduction exact quotient := div(256, 20)

Fermat original modular exponent bit2 squared base reduction exact quotient = 64/5

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(12, 20)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 240

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(256, 240)

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

Fermat original modular exponent bit2 squared base reduction:signed integer value=256;positive modulus=20;exact quotient=64/5;floor q=12

value=q*modulus+remainder;normalized remainder=16 in[0,20)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[12, 16, 4];low bit=0

next[accumulator,base,exponent]=[12, 16, 2];split exponent=2*q+bit preserves original-power invariant

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, 20)

Fermat original modular exponent bit3 squared base reduction exact quotient = 64/5

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(12, 20)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 240

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(256, 240)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 16

Fermat original modular exponent bit3 squared base reduction:signed integer value=256;positive modulus=20;exact quotient=64/5;floor q=12

value=q*modulus+remainder;normalized remainder=16 in[0,20)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[12, 16, 2];low bit=0

next[accumulator,base,exponent]=[12, 16, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(12, 16)

Fermat original modular exponent bit4 accumulator product = 192

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(192, 20)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 48/5

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(9, 20)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 180

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(192, 180)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 12

Fermat original modular exponent bit4 accumulator reduction:signed integer value=192;positive modulus=20;exact quotient=48/5;floor q=9

value=q*modulus+remainder;normalized remainder=12 in[0,20)

Fermat original modular exponent bit4 current base square := mul(16, 16)

Fermat original modular exponent bit4 current base square = 256

Fermat original modular exponent bit4 squared base reduction exact quotient := div(256, 20)

Fermat original modular exponent bit4 squared base reduction exact quotient = 64/5

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(12, 20)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 240

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(256, 240)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 16

Fermat original modular exponent bit4 squared base reduction:signed integer value=256;positive modulus=20;exact quotient=64/5;floor q=12

value=q*modulus+remainder;normalized remainder=16 in[0,20)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[12, 16, 1];low bit=1

next[accumulator,base,exponent]=[12, 16, 0];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent:remaining exponent0;final normalized accumulator=12

EXACT original base^19 modulo20=12

ACTUAL modulus=20;ORIGINAL signed base=18;prime=否;coprime=否;actual power residue=12

prime-coprime applicability=否;failed hypotheses do NOT force actual residue!=1;one successful residue does NOT prove prime

逐步反向核對

• 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 — 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• 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 accumulator 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 accumulator 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 accumulator 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 accumulator product，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

(20, 18, 否, 否, 否, 12, coprime_form_not_applicable_without_both_hypotheses)

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。