第 19841 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-converse-counterexample；資料={"modulus":561,"a":29}。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-converse-counterexample；資料={"modulus":561,"a":29}。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-converse-counterexample;original snapshot and mathematics retained

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

ACTUAL integer modulus=561;complete trial range2..floor_sqrt=23

checked rows[divisor,quotient,remainder]=[[2, 280, 1], [3, 187, 0]];first exact factor=3;prime=否

original signed base=29;positive modulus=561;use nonnegative gcd inputs=[29, 561]

ALL Euclidean rows[left,right,q,r]=[[29, 561, 0, 29], [561, 29, 19, 10], [29, 10, 2, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(29, 561)

Fermat original signed base exact quotient = 29/561

Fermat original signed base floor quotient multiple := mul(0, 561)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(29, 0)

Fermat original signed base normalized remainder = 29

Fermat original signed base:signed integer value=29;positive modulus=561;exact quotient=29/561;floor q=0

value=q*modulus+remainder;normalized remainder=29 in[0,561)

Fermat original modular exponent:original normalized base=29;original exponent=560;modulus=561

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(29, 29)

Fermat original modular exponent bit0 current base square = 841

Fermat original modular exponent bit0 squared base reduction exact quotient := div(841, 561)

Fermat original modular exponent bit0 squared base reduction exact quotient = 841/561

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(1, 561)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 561

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(841, 561)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 280

Fermat original modular exponent bit0 squared base reduction:signed integer value=841;positive modulus=561;exact quotient=841/561;floor q=1

value=q*modulus+remainder;normalized remainder=280 in[0,561)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 29, 560];low bit=0

next[accumulator,base,exponent]=[1, 280, 280];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 current base square := mul(280, 280)

Fermat original modular exponent bit1 current base square = 78400

Fermat original modular exponent bit1 squared base reduction exact quotient := div(78400, 561)

Fermat original modular exponent bit1 squared base reduction exact quotient = 78400/561

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(139, 561)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 77979

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(78400, 77979)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 421

Fermat original modular exponent bit1 squared base reduction:signed integer value=78400;positive modulus=561;exact quotient=78400/561;floor q=139

value=q*modulus+remainder;normalized remainder=421 in[0,561)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 280, 280];low bit=0

next[accumulator,base,exponent]=[1, 421, 140];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(421, 421)

Fermat original modular exponent bit2 current base square = 177241

Fermat original modular exponent bit2 squared base reduction exact quotient := div(177241, 561)

Fermat original modular exponent bit2 squared base reduction exact quotient = 177241/561

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(315, 561)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 176715

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(177241, 176715)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 526

Fermat original modular exponent bit2 squared base reduction:signed integer value=177241;positive modulus=561;exact quotient=177241/561;floor q=315

value=q*modulus+remainder;normalized remainder=526 in[0,561)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 421, 140];low bit=0

next[accumulator,base,exponent]=[1, 526, 70];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(526, 526)

Fermat original modular exponent bit3 current base square = 276676

Fermat original modular exponent bit3 squared base reduction exact quotient := div(276676, 561)

Fermat original modular exponent bit3 squared base reduction exact quotient = 276676/561

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(493, 561)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 276573

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(276676, 276573)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 103

Fermat original modular exponent bit3 squared base reduction:signed integer value=276676;positive modulus=561;exact quotient=276676/561;floor q=493

value=q*modulus+remainder;normalized remainder=103 in[0,561)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 526, 70];low bit=0

next[accumulator,base,exponent]=[1, 103, 35];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(1, 103)

Fermat original modular exponent bit4 accumulator product = 103

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(103, 561)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 103/561

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(0, 561)

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit4 accumulator reduction normalized remainder := sub(103, 0)

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 103

Fermat original modular exponent bit4 accumulator reduction:signed integer value=103;positive modulus=561;exact quotient=103/561;floor q=0

value=q*modulus+remainder;normalized remainder=103 in[0,561)

Fermat original modular exponent bit4 current base square := mul(103, 103)

Fermat original modular exponent bit4 current base square = 10609

Fermat original modular exponent bit4 squared base reduction exact quotient := div(10609, 561)

Fermat original modular exponent bit4 squared base reduction exact quotient = 10609/561

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(18, 561)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 10098

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(10609, 10098)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 511

Fermat original modular exponent bit4 squared base reduction:signed integer value=10609;positive modulus=561;exact quotient=10609/561;floor q=18

value=q*modulus+remainder;normalized remainder=511 in[0,561)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[1, 103, 35];low bit=1

next[accumulator,base,exponent]=[103, 511, 17];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 accumulator product := mul(103, 511)

Fermat original modular exponent bit5 accumulator product = 52633

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(52633, 561)

Fermat original modular exponent bit5 accumulator reduction exact quotient = 52633/561

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(93, 561)

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 52173

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(52633, 52173)

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 460

Fermat original modular exponent bit5 accumulator reduction:signed integer value=52633;positive modulus=561;exact quotient=52633/561;floor q=93

value=q*modulus+remainder;normalized remainder=460 in[0,561)

Fermat original modular exponent bit5 current base square := mul(511, 511)

Fermat original modular exponent bit5 current base square = 261121

Fermat original modular exponent bit5 squared base reduction exact quotient := div(261121, 561)

Fermat original modular exponent bit5 squared base reduction exact quotient = 261121/561

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(465, 561)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 260865

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(261121, 260865)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 256

Fermat original modular exponent bit5 squared base reduction:signed integer value=261121;positive modulus=561;exact quotient=261121/561;floor q=465

value=q*modulus+remainder;normalized remainder=256 in[0,561)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[103, 511, 17];low bit=1

next[accumulator,base,exponent]=[460, 256, 8];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit6 current base square := mul(256, 256)

Fermat original modular exponent bit6 current base square = 65536

Fermat original modular exponent bit6 squared base reduction exact quotient := div(65536, 561)

Fermat original modular exponent bit6 squared base reduction exact quotient = 65536/561

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(116, 561)

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 65076

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(65536, 65076)

Fermat original modular exponent bit6 squared base reduction normalized remainder = 460

Fermat original modular exponent bit6 squared base reduction:signed integer value=65536;positive modulus=561;exact quotient=65536/561;floor q=116

value=q*modulus+remainder;normalized remainder=460 in[0,561)

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[460, 256, 8];low bit=0

next[accumulator,base,exponent]=[460, 460, 4];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit7 current base square := mul(460, 460)

Fermat original modular exponent bit7 current base square = 211600

Fermat original modular exponent bit7 squared base reduction exact quotient := div(211600, 561)

Fermat original modular exponent bit7 squared base reduction exact quotient = 211600/561

Fermat original modular exponent bit7 squared base reduction floor quotient multiple := mul(377, 561)

Fermat original modular exponent bit7 squared base reduction floor quotient multiple = 211497

Fermat original modular exponent bit7 squared base reduction normalized remainder := sub(211600, 211497)

Fermat original modular exponent bit7 squared base reduction normalized remainder = 103

Fermat original modular exponent bit7 squared base reduction:signed integer value=211600;positive modulus=561;exact quotient=211600/561;floor q=377

value=q*modulus+remainder;normalized remainder=103 in[0,561)

Fermat original modular exponent:step=7;previous[accumulator,base,exponent]=[460, 460, 4];low bit=0

next[accumulator,base,exponent]=[460, 103, 2];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit8 current base square := mul(103, 103)

Fermat original modular exponent bit8 current base square = 10609

Fermat original modular exponent bit8 squared base reduction exact quotient := div(10609, 561)

Fermat original modular exponent bit8 squared base reduction exact quotient = 10609/561

Fermat original modular exponent bit8 squared base reduction floor quotient multiple := mul(18, 561)

Fermat original modular exponent bit8 squared base reduction floor quotient multiple = 10098

Fermat original modular exponent bit8 squared base reduction normalized remainder := sub(10609, 10098)

Fermat original modular exponent bit8 squared base reduction normalized remainder = 511

Fermat original modular exponent bit8 squared base reduction:signed integer value=10609;positive modulus=561;exact quotient=10609/561;floor q=18

value=q*modulus+remainder;normalized remainder=511 in[0,561)

Fermat original modular exponent:step=8;previous[accumulator,base,exponent]=[460, 103, 2];low bit=0

next[accumulator,base,exponent]=[460, 511, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit9 accumulator product := mul(460, 511)

Fermat original modular exponent bit9 accumulator product = 235060

Fermat original modular exponent bit9 accumulator reduction exact quotient := div(235060, 561)

Fermat original modular exponent bit9 accumulator reduction exact quotient = 235060/561

Fermat original modular exponent bit9 accumulator reduction floor quotient multiple := mul(419, 561)

Fermat original modular exponent bit9 accumulator reduction floor quotient multiple = 235059

Fermat original modular exponent bit9 accumulator reduction normalized remainder := sub(235060, 235059)

Fermat original modular exponent bit9 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit9 accumulator reduction:signed integer value=235060;positive modulus=561;exact quotient=235060/561;floor q=419

value=q*modulus+remainder;normalized remainder=1 in[0,561)

Fermat original modular exponent bit9 current base square := mul(511, 511)

Fermat original modular exponent bit9 current base square = 261121

Fermat original modular exponent bit9 squared base reduction exact quotient := div(261121, 561)

Fermat original modular exponent bit9 squared base reduction exact quotient = 261121/561

Fermat original modular exponent bit9 squared base reduction floor quotient multiple := mul(465, 561)

Fermat original modular exponent bit9 squared base reduction floor quotient multiple = 260865

Fermat original modular exponent bit9 squared base reduction normalized remainder := sub(261121, 260865)

Fermat original modular exponent bit9 squared base reduction normalized remainder = 256

Fermat original modular exponent bit9 squared base reduction:signed integer value=261121;positive modulus=561;exact quotient=261121/561;floor q=465

value=q*modulus+remainder;normalized remainder=256 in[0,561)

Fermat original modular exponent:step=9;previous[accumulator,base,exponent]=[460, 511, 1];low bit=1

next[accumulator,base,exponent]=[1, 256, 0];split exponent=2*q+bit preserves original-power invariant

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

EXACT original base^560 modulo561=1

ACTUAL composite modulus=561 certified by a trial factor;ORIGINAL base=29 coprime;computed a^(m-1) modulo m=1

THIS chosen composite/base pair passes the Fermat power test,but modulus is NOT prime;counterexample to converse ONLY

逐步反向核對

• S085 · 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。
• S084 · 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。
• S083 · 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。
• S082 · 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。
• S081 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 squared base reduction normalized remainder，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 squared base reduction floor quotient multiple，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 squared base reduction exact quotient，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 current base square，不以小數近似替代。
• S077 · 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。
• S076 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 accumulator reduction normalized remainder，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 accumulator reduction floor quotient multiple，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 accumulator reduction exact quotient，不以小數近似替代。
• S073 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit9 accumulator product，不以小數近似替代。
• 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 — 以精確有理數重算 Fermat original modular exponent bit8 squared base reduction normalized remainder，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit8 squared base reduction floor quotient multiple，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit8 squared base reduction exact quotient，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit8 current base square，不以小數近似替代。
• S066 · 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。
• 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 — 以精確有理數重算 Fermat original modular exponent bit7 squared base reduction normalized remainder，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit7 squared base reduction floor quotient multiple，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit7 squared base reduction exact quotient，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 bit7 current base square，不以小數近似替代。
• 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 original modular exponent bit6 squared base reduction normalized remainder，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 squared base reduction 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 original modular exponent bit6 squared base reduction exact quotient，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 bit6 current base square，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 — 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• 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 — 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• 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 squared base 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 squared base 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 squared base 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 current base square，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 — 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• 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 — 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• 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 squared base 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 squared base 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 squared base 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 current base square，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

(561, 29, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)

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。