第 19816 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-all-integer；資料={"p":53,"a":-848}。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-all-integer；資料={"p":53,"a":-848}。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-all-integer;original snapshot and mathematics retained

explicit ordered source output fields=[p, normalized_base, power_p, comparison_residue, congruent, boundary]

ACTUAL integer modulus=53;complete trial range2..floor_sqrt=7

checked rows[divisor,quotient,remainder]=[[2, 26, 1], [3, 17, 2], [4, 13, 1], [5, 10, 3], [6, 8, 5], [7, 7, 4]];first exact factor=null;prime=是

original signed base=-848;positive modulus=53;use nonnegative gcd inputs=[848, 53]

ALL Euclidean rows[left,right,q,r]=[[848, 53, 16, 0]];terminal nonnegative gcd=53;coprime=否

Fermat original signed base exact quotient := div(-848, 53)

Fermat original signed base exact quotient = -16

Fermat original signed base floor quotient multiple := mul(-16, 53)

Fermat original signed base floor quotient multiple = -848

Fermat original signed base normalized remainder := sub(-848, -848)

Fermat original signed base normalized remainder = 0

Fermat original signed base:signed integer value=-848;positive modulus=53;exact quotient=-16;floor q=-16

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent:original normalized base=0;original exponent=53;modulus=53

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, 0)

Fermat original modular exponent bit0 accumulator product = 0

Fermat original modular exponent bit0 accumulator reduction exact quotient := div(0, 53)

Fermat original modular exponent bit0 accumulator reduction exact quotient = 0

Fermat original modular exponent bit0 accumulator reduction floor quotient multiple := mul(0, 53)

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

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

Fermat original modular exponent bit0 accumulator reduction normalized remainder = 0

Fermat original modular exponent bit0 accumulator reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent bit0 current base square := mul(0, 0)

Fermat original modular exponent bit0 current base square = 0

Fermat original modular exponent bit0 squared base reduction exact quotient := div(0, 53)

Fermat original modular exponent bit0 squared base reduction exact quotient = 0

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(0, 53)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(0, 0)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 0

Fermat original modular exponent bit0 squared base reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 0, 53];low bit=1

next[accumulator,base,exponent]=[0, 0, 26];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 current base square := mul(0, 0)

Fermat original modular exponent bit1 current base square = 0

Fermat original modular exponent bit1 squared base reduction exact quotient := div(0, 53)

Fermat original modular exponent bit1 squared base reduction exact quotient = 0

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(0, 53)

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

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

Fermat original modular exponent bit1 squared base reduction normalized remainder = 0

Fermat original modular exponent bit1 squared base reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[0, 0, 26];low bit=0

next[accumulator,base,exponent]=[0, 0, 13];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 accumulator product := mul(0, 0)

Fermat original modular exponent bit2 accumulator product = 0

Fermat original modular exponent bit2 accumulator reduction exact quotient := div(0, 53)

Fermat original modular exponent bit2 accumulator reduction exact quotient = 0

Fermat original modular exponent bit2 accumulator reduction floor quotient multiple := mul(0, 53)

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

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

Fermat original modular exponent bit2 accumulator reduction normalized remainder = 0

Fermat original modular exponent bit2 accumulator reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent bit2 current base square := mul(0, 0)

Fermat original modular exponent bit2 current base square = 0

Fermat original modular exponent bit2 squared base reduction exact quotient := div(0, 53)

Fermat original modular exponent bit2 squared base reduction exact quotient = 0

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(0, 53)

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

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

Fermat original modular exponent bit2 squared base reduction normalized remainder = 0

Fermat original modular exponent bit2 squared base reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[0, 0, 13];low bit=1

next[accumulator,base,exponent]=[0, 0, 6];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(0, 0)

Fermat original modular exponent bit3 current base square = 0

Fermat original modular exponent bit3 squared base reduction exact quotient := div(0, 53)

Fermat original modular exponent bit3 squared base reduction exact quotient = 0

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(0, 53)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(0, 0)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 0

Fermat original modular exponent bit3 squared base reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[0, 0, 6];low bit=0

next[accumulator,base,exponent]=[0, 0, 3];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 accumulator product := mul(0, 0)

Fermat original modular exponent bit4 accumulator product = 0

Fermat original modular exponent bit4 accumulator reduction exact quotient := div(0, 53)

Fermat original modular exponent bit4 accumulator reduction exact quotient = 0

Fermat original modular exponent bit4 accumulator reduction floor quotient multiple := mul(0, 53)

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

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

Fermat original modular exponent bit4 accumulator reduction normalized remainder = 0

Fermat original modular exponent bit4 accumulator reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent bit4 current base square := mul(0, 0)

Fermat original modular exponent bit4 current base square = 0

Fermat original modular exponent bit4 squared base reduction exact quotient := div(0, 53)

Fermat original modular exponent bit4 squared base reduction exact quotient = 0

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(0, 53)

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

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

Fermat original modular exponent bit4 squared base reduction normalized remainder = 0

Fermat original modular exponent bit4 squared base reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[0, 0, 3];low bit=1

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

Fermat original modular exponent bit5 accumulator product := mul(0, 0)

Fermat original modular exponent bit5 accumulator product = 0

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(0, 53)

Fermat original modular exponent bit5 accumulator reduction exact quotient = 0

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(0, 53)

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(0, 0)

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 0

Fermat original modular exponent bit5 accumulator reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent bit5 current base square := mul(0, 0)

Fermat original modular exponent bit5 current base square = 0

Fermat original modular exponent bit5 squared base reduction exact quotient := div(0, 53)

Fermat original modular exponent bit5 squared base reduction exact quotient = 0

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(0, 53)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(0, 0)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 0

Fermat original modular exponent bit5 squared base reduction:signed integer value=0;positive modulus=53;exact quotient=0;floor q=0

value=q*modulus+remainder;normalized remainder=0 in[0,53)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[0, 0, 1];low bit=1

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

Fermat original modular exponent:remaining exponent0;final normalized accumulator=0

EXACT original base^53 modulo53=0

ACTUAL prime p=53;original a=-848 is divisible by p;normalized residue0;positive exponent p

a^p residue=0=0=a residue;NO cancellation of a or0 needed

逐步反向核對

• 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 — 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 — 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 — 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• 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 — 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

(53, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)

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。