第 20040 題：Euler定理

原題與已知條件

n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":19,"a":-16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-negative-base-normalization；資料={"n":19,"a":-16}。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-euler-theorem-number-theory

逐步推導

source operation=eulerthm-negative-base-normalization;original snapshot and mathematics retained

explicit ordered source output fields=[original_base, normalized_base, totient, residue, congruent_to_one, certificate]

Euler ORIGINAL signed base normalization exact quotient := div(-16, 19)

Euler ORIGINAL signed base normalization exact quotient = -16/19

Euler ORIGINAL signed base normalization floor quotient multiple := mul(-1, 19)

Euler ORIGINAL signed base normalization floor quotient multiple = -19

Euler ORIGINAL signed base normalization normalized remainder := sub(-16, -19)

Euler ORIGINAL signed base normalization normalized remainder = 3

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=-16;modulus=19;exact ratio=-16/19;floor quotient=-1

value=q*n+remainder;normalized residue=3 in[0,19);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[-16, 19];nonnegative initial pair=[16, 19]

ALL exact Euclidean rows[left,right,q,r]=[[16, 19, 0, 16], [19, 16, 1, 3], [16, 3, 5, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 19];nonnegative initial pair=[0, 19]

ALL exact Euclidean rows[left,right,q,r]=[[0, 19, 0, 0]];terminal gcd=19

Euler residue representative1:ORIGINAL gcd inputs=[1, 19];nonnegative initial pair=[1, 19]

ALL exact Euclidean rows[left,right,q,r]=[[1, 19, 0, 1], [19, 1, 19, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 19];nonnegative initial pair=[2, 19]

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

Euler residue representative3:ORIGINAL gcd inputs=[3, 19];nonnegative initial pair=[3, 19]

ALL exact Euclidean rows[left,right,q,r]=[[3, 19, 0, 3], [19, 3, 6, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 19];nonnegative initial pair=[4, 19]

ALL exact Euclidean rows[left,right,q,r]=[[4, 19, 0, 4], [19, 4, 4, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 19];nonnegative initial pair=[5, 19]

ALL exact Euclidean rows[left,right,q,r]=[[5, 19, 0, 5], [19, 5, 3, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 19];nonnegative initial pair=[6, 19]

ALL exact Euclidean rows[left,right,q,r]=[[6, 19, 0, 6], [19, 6, 3, 1], [6, 1, 6, 0]];terminal gcd=1

Euler residue representative7:ORIGINAL gcd inputs=[7, 19];nonnegative initial pair=[7, 19]

ALL exact Euclidean rows[left,right,q,r]=[[7, 19, 0, 7], [19, 7, 2, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 19];nonnegative initial pair=[8, 19]

ALL exact Euclidean rows[left,right,q,r]=[[8, 19, 0, 8], [19, 8, 2, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 19];nonnegative initial pair=[9, 19]

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

Euler residue representative10:ORIGINAL gcd inputs=[10, 19];nonnegative initial pair=[10, 19]

ALL exact Euclidean rows[left,right,q,r]=[[10, 19, 0, 10], [19, 10, 1, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal gcd=1

Euler residue representative11:ORIGINAL gcd inputs=[11, 19];nonnegative initial pair=[11, 19]

ALL exact Euclidean rows[left,right,q,r]=[[11, 19, 0, 11], [19, 11, 1, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 19];nonnegative initial pair=[12, 19]

ALL exact Euclidean rows[left,right,q,r]=[[12, 19, 0, 12], [19, 12, 1, 7], [12, 7, 1, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative13:ORIGINAL gcd inputs=[13, 19];nonnegative initial pair=[13, 19]

ALL exact Euclidean rows[left,right,q,r]=[[13, 19, 0, 13], [19, 13, 1, 6], [13, 6, 2, 1], [6, 1, 6, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 19];nonnegative initial pair=[14, 19]

ALL exact Euclidean rows[left,right,q,r]=[[14, 19, 0, 14], [19, 14, 1, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative15:ORIGINAL gcd inputs=[15, 19];nonnegative initial pair=[15, 19]

ALL exact Euclidean rows[left,right,q,r]=[[15, 19, 0, 15], [19, 15, 1, 4], [15, 4, 3, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative16:ORIGINAL gcd inputs=[16, 19];nonnegative initial pair=[16, 19]

ALL exact Euclidean rows[left,right,q,r]=[[16, 19, 0, 16], [19, 16, 1, 3], [16, 3, 5, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative17:ORIGINAL gcd inputs=[17, 19];nonnegative initial pair=[17, 19]

ALL exact Euclidean rows[left,right,q,r]=[[17, 19, 0, 17], [19, 17, 1, 2], [17, 2, 8, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 19];nonnegative initial pair=[18, 19]

ALL exact Euclidean rows[left,right,q,r]=[[18, 19, 0, 18], [19, 18, 1, 1], [18, 1, 18, 0]];terminal gcd=1

Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Euler complete reduced residue count phi = 18

COMPLETE original residue carrier0..18;ALL [representative,gcd,unit] rows=[[0, 19, 否], [1, 1, 是], [2, 1, 是], [3, 1, 是], [4, 1, 是], [5, 1, 是], [6, 1, 是], [7, 1, 是], [8, 1, 是], [9, 1, 是], [10, 1, 是], [11, 1, 是], [12, 1, 是], [13, 1, 是], [14, 1, 是], [15, 1, 是], [16, 1, 是], [17, 1, 是], [18, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];phi(n)=18;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

ORIGINAL normalized base=3;COMPLETE totient exponent=18;modulus=19

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

Euler power bit0 base square := mul(3, 3)

Euler power bit0 base square = 9

Euler power bit0 squared-base reduction exact quotient := div(9, 19)

Euler power bit0 squared-base reduction exact quotient = 9/19

Euler power bit0 squared-base reduction floor quotient multiple := mul(0, 19)

Euler power bit0 squared-base reduction floor quotient multiple = 0

Euler power bit0 squared-base reduction normalized remainder := sub(9, 0)

Euler power bit0 squared-base reduction normalized remainder = 9

Euler power bit0 squared-base reduction:ORIGINAL signed integer=9;modulus=19;exact ratio=9/19;floor quotient=0

value=q*n+remainder;normalized residue=9 in[0,19);same congruence class,NOT absolute-value replacement

step=0;previous[accumulator,base,exponent]=[1, 3, 18];low bit=0

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

Euler power bit1 accumulator product := mul(1, 9)

Euler power bit1 accumulator product = 9

Euler power bit1 accumulator reduction exact quotient := div(9, 19)

Euler power bit1 accumulator reduction exact quotient = 9/19

Euler power bit1 accumulator reduction floor quotient multiple := mul(0, 19)

Euler power bit1 accumulator reduction floor quotient multiple = 0

Euler power bit1 accumulator reduction normalized remainder := sub(9, 0)

Euler power bit1 accumulator reduction normalized remainder = 9

Euler power bit1 accumulator reduction:ORIGINAL signed integer=9;modulus=19;exact ratio=9/19;floor quotient=0

value=q*n+remainder;normalized residue=9 in[0,19);same congruence class,NOT absolute-value replacement

Euler power bit1 base square := mul(9, 9)

Euler power bit1 base square = 81

Euler power bit1 squared-base reduction exact quotient := div(81, 19)

Euler power bit1 squared-base reduction exact quotient = 81/19

Euler power bit1 squared-base reduction floor quotient multiple := mul(4, 19)

Euler power bit1 squared-base reduction floor quotient multiple = 76

Euler power bit1 squared-base reduction normalized remainder := sub(81, 76)

Euler power bit1 squared-base reduction normalized remainder = 5

Euler power bit1 squared-base reduction:ORIGINAL signed integer=81;modulus=19;exact ratio=81/19;floor quotient=4

value=q*n+remainder;normalized residue=5 in[0,19);same congruence class,NOT absolute-value replacement

step=1;previous[accumulator,base,exponent]=[1, 9, 9];low bit=1

next[accumulator,base,exponent]=[9, 5, 4];split exponent=2*q+bit preserves original-power invariant

Euler power bit2 base square := mul(5, 5)

Euler power bit2 base square = 25

Euler power bit2 squared-base reduction exact quotient := div(25, 19)

Euler power bit2 squared-base reduction exact quotient = 25/19

Euler power bit2 squared-base reduction floor quotient multiple := mul(1, 19)

Euler power bit2 squared-base reduction floor quotient multiple = 19

Euler power bit2 squared-base reduction normalized remainder := sub(25, 19)

Euler power bit2 squared-base reduction normalized remainder = 6

Euler power bit2 squared-base reduction:ORIGINAL signed integer=25;modulus=19;exact ratio=25/19;floor quotient=1

value=q*n+remainder;normalized residue=6 in[0,19);same congruence class,NOT absolute-value replacement

step=2;previous[accumulator,base,exponent]=[9, 5, 4];low bit=0

next[accumulator,base,exponent]=[9, 6, 2];split exponent=2*q+bit preserves original-power invariant

Euler power bit3 base square := mul(6, 6)

Euler power bit3 base square = 36

Euler power bit3 squared-base reduction exact quotient := div(36, 19)

Euler power bit3 squared-base reduction exact quotient = 36/19

Euler power bit3 squared-base reduction floor quotient multiple := mul(1, 19)

Euler power bit3 squared-base reduction floor quotient multiple = 19

Euler power bit3 squared-base reduction normalized remainder := sub(36, 19)

Euler power bit3 squared-base reduction normalized remainder = 17

Euler power bit3 squared-base reduction:ORIGINAL signed integer=36;modulus=19;exact ratio=36/19;floor quotient=1

value=q*n+remainder;normalized residue=17 in[0,19);same congruence class,NOT absolute-value replacement

step=3;previous[accumulator,base,exponent]=[9, 6, 2];low bit=0

next[accumulator,base,exponent]=[9, 17, 1];split exponent=2*q+bit preserves original-power invariant

Euler power bit4 accumulator product := mul(9, 17)

Euler power bit4 accumulator product = 153

Euler power bit4 accumulator reduction exact quotient := div(153, 19)

Euler power bit4 accumulator reduction exact quotient = 153/19

Euler power bit4 accumulator reduction floor quotient multiple := mul(8, 19)

Euler power bit4 accumulator reduction floor quotient multiple = 152

Euler power bit4 accumulator reduction normalized remainder := sub(153, 152)

Euler power bit4 accumulator reduction normalized remainder = 1

Euler power bit4 accumulator reduction:ORIGINAL signed integer=153;modulus=19;exact ratio=153/19;floor quotient=8

value=q*n+remainder;normalized residue=1 in[0,19);same congruence class,NOT absolute-value replacement

Euler power bit4 base square := mul(17, 17)

Euler power bit4 base square = 289

Euler power bit4 squared-base reduction exact quotient := div(289, 19)

Euler power bit4 squared-base reduction exact quotient = 289/19

Euler power bit4 squared-base reduction floor quotient multiple := mul(15, 19)

Euler power bit4 squared-base reduction floor quotient multiple = 285

Euler power bit4 squared-base reduction normalized remainder := sub(289, 285)

Euler power bit4 squared-base reduction normalized remainder = 4

Euler power bit4 squared-base reduction:ORIGINAL signed integer=289;modulus=19;exact ratio=289/19;floor quotient=15

value=q*n+remainder;normalized residue=4 in[0,19);same congruence class,NOT absolute-value replacement

step=4;previous[accumulator,base,exponent]=[9, 17, 1];low bit=1

next[accumulator,base,exponent]=[1, 4, 0];split exponent=2*q+bit preserves original-power invariant

remaining exponent0;final normalized accumulator=1

EXACT original normalized base^18 modulo19=1

Euler ACTUAL base-unit inverse product := mul(3, 13)

Euler ACTUAL base-unit inverse product = 39

Euler ACTUAL inverse check exact quotient := div(39, 19)

Euler ACTUAL inverse check exact quotient = 39/19

Euler ACTUAL inverse check floor quotient multiple := mul(2, 19)

Euler ACTUAL inverse check floor quotient multiple = 38

Euler ACTUAL inverse check normalized remainder := sub(39, 38)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=39;modulus=19;exact ratio=39/19;floor quotient=2

value=q*n+remainder;normalized residue=1 in[0,19);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=19;gcd(original a=-16,n)=1;normalized base=3;ACTUAL inverse=13;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

for ANY r in U(n),a*r remains a unit;actual inverse makes multiplication injective;finite self-injection on U(n) is surjective,so a*U(n)=U(n)

SAME complete U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] of size phi=18;multiplication by original a=-16 permutes it

multiply ALL unit images:a^(18)*product(U) congruent product(U);EVERY U factor is a UNIT,so product(U) has reverse-product inverse;cancel ONLY this unit to get a^phi congruent1 modulo19

逐步反向核對

• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。
• S072 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check floor quotient multiple，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL inverse check exact quotient，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Euler ACTUAL base-unit inverse product，不以小數近似替代。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 squared-base reduction normalized remainder，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 squared-base reduction floor quotient multiple，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 squared-base reduction exact quotient，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 base square，不以小數近似替代。
• S062 · 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。
• S061 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 accumulator reduction normalized remainder，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 accumulator reduction floor quotient multiple，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 accumulator reduction exact quotient，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Euler power bit4 accumulator product，不以小數近似替代。
• S057 · 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。
• 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 — 以精確有理數重算 Euler power bit3 squared-base reduction normalized remainder，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction floor quotient multiple，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 squared-base reduction exact quotient，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 base square，不以小數近似替代。
• S051 · 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。
• S050 · 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。
• S049 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction normalized remainder，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction floor quotient multiple，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 squared-base reduction exact quotient，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 base square，不以小數近似替代。
• S045 · 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。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 squared-base reduction normalized remainder，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 squared-base reduction floor quotient multiple，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 squared-base reduction exact quotient，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 base square，不以小數近似替代。
• 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 — 以精確有理數重算 Euler power bit1 accumulator reduction normalized remainder，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 accumulator reduction floor quotient multiple，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 accumulator reduction exact quotient，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Euler power bit1 accumulator product，不以小數近似替代。
• S034 · 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。
• S033 · 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。
• S032 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction normalized remainder，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction floor quotient multiple，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 squared-base reduction exact quotient，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Euler power bit0 base square，不以小數近似替代。
• S028 · 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。
• S027 · 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。
• S026 · premise_and_result_audit — 以精確有理數重算 Euler complete reduced residue count phi，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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · 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。
• S021 · 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。
• 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 — 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · 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。
• S015 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · 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。
• S004 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization normalized remainder，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization floor quotient multiple，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Euler ORIGINAL signed base normalization exact quotient，不以小數近似替代。
• 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

(-16, 3, 18, 1, 是, negative_base_normalized_before_exact_modular_power)

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。