第 20050 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":29,"a":-26}。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":29,"a":-26}。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(-26, 29)

Euler ORIGINAL signed base normalization exact quotient = -26/29

Euler ORIGINAL signed base normalization floor quotient multiple := mul(-1, 29)

Euler ORIGINAL signed base normalization floor quotient multiple = -29

Euler ORIGINAL signed base normalization normalized remainder := sub(-26, -29)

Euler ORIGINAL signed base normalization normalized remainder = 3

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=-26;modulus=29;exact ratio=-26/29;floor quotient=-1

value=q*n+remainder;normalized residue=3 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[-26, 29];nonnegative initial pair=[26, 29]

ALL exact Euclidean rows[left,right,q,r]=[[26, 29, 0, 26], [29, 26, 1, 3], [26, 3, 8, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative0:ORIGINAL gcd inputs=[0, 29];nonnegative initial pair=[0, 29]

ALL exact Euclidean rows[left,right,q,r]=[[0, 29, 0, 0]];terminal gcd=29

Euler residue representative1:ORIGINAL gcd inputs=[1, 29];nonnegative initial pair=[1, 29]

ALL exact Euclidean rows[left,right,q,r]=[[1, 29, 0, 1], [29, 1, 29, 0]];terminal gcd=1

Euler residue representative2:ORIGINAL gcd inputs=[2, 29];nonnegative initial pair=[2, 29]

ALL exact Euclidean rows[left,right,q,r]=[[2, 29, 0, 2], [29, 2, 14, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative3:ORIGINAL gcd inputs=[3, 29];nonnegative initial pair=[3, 29]

ALL exact Euclidean rows[left,right,q,r]=[[3, 29, 0, 3], [29, 3, 9, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative4:ORIGINAL gcd inputs=[4, 29];nonnegative initial pair=[4, 29]

ALL exact Euclidean rows[left,right,q,r]=[[4, 29, 0, 4], [29, 4, 7, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative5:ORIGINAL gcd inputs=[5, 29];nonnegative initial pair=[5, 29]

ALL exact Euclidean rows[left,right,q,r]=[[5, 29, 0, 5], [29, 5, 5, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative6:ORIGINAL gcd inputs=[6, 29];nonnegative initial pair=[6, 29]

ALL exact Euclidean rows[left,right,q,r]=[[6, 29, 0, 6], [29, 6, 4, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative7:ORIGINAL gcd inputs=[7, 29];nonnegative initial pair=[7, 29]

ALL exact Euclidean rows[left,right,q,r]=[[7, 29, 0, 7], [29, 7, 4, 1], [7, 1, 7, 0]];terminal gcd=1

Euler residue representative8:ORIGINAL gcd inputs=[8, 29];nonnegative initial pair=[8, 29]

ALL exact Euclidean rows[left,right,q,r]=[[8, 29, 0, 8], [29, 8, 3, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative9:ORIGINAL gcd inputs=[9, 29];nonnegative initial pair=[9, 29]

ALL exact Euclidean rows[left,right,q,r]=[[9, 29, 0, 9], [29, 9, 3, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative10:ORIGINAL gcd inputs=[10, 29];nonnegative initial pair=[10, 29]

ALL exact Euclidean rows[left,right,q,r]=[[10, 29, 0, 10], [29, 10, 2, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal gcd=1

Euler residue representative11:ORIGINAL gcd inputs=[11, 29];nonnegative initial pair=[11, 29]

ALL exact Euclidean rows[left,right,q,r]=[[11, 29, 0, 11], [29, 11, 2, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative12:ORIGINAL gcd inputs=[12, 29];nonnegative initial pair=[12, 29]

ALL exact Euclidean rows[left,right,q,r]=[[12, 29, 0, 12], [29, 12, 2, 5], [12, 5, 2, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative13:ORIGINAL gcd inputs=[13, 29];nonnegative initial pair=[13, 29]

ALL exact Euclidean rows[left,right,q,r]=[[13, 29, 0, 13], [29, 13, 2, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative14:ORIGINAL gcd inputs=[14, 29];nonnegative initial pair=[14, 29]

ALL exact Euclidean rows[left,right,q,r]=[[14, 29, 0, 14], [29, 14, 2, 1], [14, 1, 14, 0]];terminal gcd=1

Euler residue representative15:ORIGINAL gcd inputs=[15, 29];nonnegative initial pair=[15, 29]

ALL exact Euclidean rows[left,right,q,r]=[[15, 29, 0, 15], [29, 15, 1, 14], [15, 14, 1, 1], [14, 1, 14, 0]];terminal gcd=1

Euler residue representative16:ORIGINAL gcd inputs=[16, 29];nonnegative initial pair=[16, 29]

ALL exact Euclidean rows[left,right,q,r]=[[16, 29, 0, 16], [29, 16, 1, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative17:ORIGINAL gcd inputs=[17, 29];nonnegative initial pair=[17, 29]

ALL exact Euclidean rows[left,right,q,r]=[[17, 29, 0, 17], [29, 17, 1, 12], [17, 12, 1, 5], [12, 5, 2, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative18:ORIGINAL gcd inputs=[18, 29];nonnegative initial pair=[18, 29]

ALL exact Euclidean rows[left,right,q,r]=[[18, 29, 0, 18], [29, 18, 1, 11], [18, 11, 1, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Euler residue representative19:ORIGINAL gcd inputs=[19, 29];nonnegative initial pair=[19, 29]

ALL exact Euclidean rows[left,right,q,r]=[[19, 29, 0, 19], [29, 19, 1, 10], [19, 10, 1, 9], [10, 9, 1, 1], [9, 1, 9, 0]];terminal gcd=1

Euler residue representative20:ORIGINAL gcd inputs=[20, 29];nonnegative initial pair=[20, 29]

ALL exact Euclidean rows[left,right,q,r]=[[20, 29, 0, 20], [29, 20, 1, 9], [20, 9, 2, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative21:ORIGINAL gcd inputs=[21, 29];nonnegative initial pair=[21, 29]

ALL exact Euclidean rows[left,right,q,r]=[[21, 29, 0, 21], [29, 21, 1, 8], [21, 8, 2, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative22:ORIGINAL gcd inputs=[22, 29];nonnegative initial pair=[22, 29]

ALL exact Euclidean rows[left,right,q,r]=[[22, 29, 0, 22], [29, 22, 1, 7], [22, 7, 3, 1], [7, 1, 7, 0]];terminal gcd=1

Euler residue representative23:ORIGINAL gcd inputs=[23, 29];nonnegative initial pair=[23, 29]

ALL exact Euclidean rows[left,right,q,r]=[[23, 29, 0, 23], [29, 23, 1, 6], [23, 6, 3, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Euler residue representative24:ORIGINAL gcd inputs=[24, 29];nonnegative initial pair=[24, 29]

ALL exact Euclidean rows[left,right,q,r]=[[24, 29, 0, 24], [29, 24, 1, 5], [24, 5, 4, 4], [5, 4, 1, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative25:ORIGINAL gcd inputs=[25, 29];nonnegative initial pair=[25, 29]

ALL exact Euclidean rows[left,right,q,r]=[[25, 29, 0, 25], [29, 25, 1, 4], [25, 4, 6, 1], [4, 1, 4, 0]];terminal gcd=1

Euler residue representative26:ORIGINAL gcd inputs=[26, 29];nonnegative initial pair=[26, 29]

ALL exact Euclidean rows[left,right,q,r]=[[26, 29, 0, 26], [29, 26, 1, 3], [26, 3, 8, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative27:ORIGINAL gcd inputs=[27, 29];nonnegative initial pair=[27, 29]

ALL exact Euclidean rows[left,right,q,r]=[[27, 29, 0, 27], [29, 27, 1, 2], [27, 2, 13, 1], [2, 1, 2, 0]];terminal gcd=1

Euler residue representative28:ORIGINAL gcd inputs=[28, 29];nonnegative initial pair=[28, 29]

ALL exact Euclidean rows[left,right,q,r]=[[28, 29, 0, 28], [29, 28, 1, 1], [28, 1, 28, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Euler complete reduced residue count phi = 28

COMPLETE original residue carrier0..28;ALL [representative,gcd,unit] rows=[[0, 29, 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], [19, 1, 是], [20, 1, 是], [21, 1, 是], [22, 1, 是], [23, 1, 是], [24, 1, 是], [25, 1, 是], [26, 1, 是], [27, 1, 是], [28, 1, 是]]

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28];phi(n)=28;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

ORIGINAL normalized base=3;COMPLETE totient exponent=28;modulus=29

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, 29)

Euler power bit0 squared-base reduction exact quotient = 9/29

Euler power bit0 squared-base reduction floor quotient multiple := mul(0, 29)

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=29;exact ratio=9/29;floor quotient=0

value=q*n+remainder;normalized residue=9 in[0,29);same congruence class,NOT absolute-value replacement

step=0;previous[accumulator,base,exponent]=[1, 3, 28];low bit=0

next[accumulator,base,exponent]=[1, 9, 14];split exponent=2*q+bit preserves original-power invariant

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

Euler power bit1 base square = 81

Euler power bit1 squared-base reduction exact quotient := div(81, 29)

Euler power bit1 squared-base reduction exact quotient = 81/29

Euler power bit1 squared-base reduction floor quotient multiple := mul(2, 29)

Euler power bit1 squared-base reduction floor quotient multiple = 58

Euler power bit1 squared-base reduction normalized remainder := sub(81, 58)

Euler power bit1 squared-base reduction normalized remainder = 23

Euler power bit1 squared-base reduction:ORIGINAL signed integer=81;modulus=29;exact ratio=81/29;floor quotient=2

value=q*n+remainder;normalized residue=23 in[0,29);same congruence class,NOT absolute-value replacement

step=1;previous[accumulator,base,exponent]=[1, 9, 14];low bit=0

next[accumulator,base,exponent]=[1, 23, 7];split exponent=2*q+bit preserves original-power invariant

Euler power bit2 accumulator product := mul(1, 23)

Euler power bit2 accumulator product = 23

Euler power bit2 accumulator reduction exact quotient := div(23, 29)

Euler power bit2 accumulator reduction exact quotient = 23/29

Euler power bit2 accumulator reduction floor quotient multiple := mul(0, 29)

Euler power bit2 accumulator reduction floor quotient multiple = 0

Euler power bit2 accumulator reduction normalized remainder := sub(23, 0)

Euler power bit2 accumulator reduction normalized remainder = 23

Euler power bit2 accumulator reduction:ORIGINAL signed integer=23;modulus=29;exact ratio=23/29;floor quotient=0

value=q*n+remainder;normalized residue=23 in[0,29);same congruence class,NOT absolute-value replacement

Euler power bit2 base square := mul(23, 23)

Euler power bit2 base square = 529

Euler power bit2 squared-base reduction exact quotient := div(529, 29)

Euler power bit2 squared-base reduction exact quotient = 529/29

Euler power bit2 squared-base reduction floor quotient multiple := mul(18, 29)

Euler power bit2 squared-base reduction floor quotient multiple = 522

Euler power bit2 squared-base reduction normalized remainder := sub(529, 522)

Euler power bit2 squared-base reduction normalized remainder = 7

Euler power bit2 squared-base reduction:ORIGINAL signed integer=529;modulus=29;exact ratio=529/29;floor quotient=18

value=q*n+remainder;normalized residue=7 in[0,29);same congruence class,NOT absolute-value replacement

step=2;previous[accumulator,base,exponent]=[1, 23, 7];low bit=1

next[accumulator,base,exponent]=[23, 7, 3];split exponent=2*q+bit preserves original-power invariant

Euler power bit3 accumulator product := mul(23, 7)

Euler power bit3 accumulator product = 161

Euler power bit3 accumulator reduction exact quotient := div(161, 29)

Euler power bit3 accumulator reduction exact quotient = 161/29

Euler power bit3 accumulator reduction floor quotient multiple := mul(5, 29)

Euler power bit3 accumulator reduction floor quotient multiple = 145

Euler power bit3 accumulator reduction normalized remainder := sub(161, 145)

Euler power bit3 accumulator reduction normalized remainder = 16

Euler power bit3 accumulator reduction:ORIGINAL signed integer=161;modulus=29;exact ratio=161/29;floor quotient=5

value=q*n+remainder;normalized residue=16 in[0,29);same congruence class,NOT absolute-value replacement

Euler power bit3 base square := mul(7, 7)

Euler power bit3 base square = 49

Euler power bit3 squared-base reduction exact quotient := div(49, 29)

Euler power bit3 squared-base reduction exact quotient = 49/29

Euler power bit3 squared-base reduction floor quotient multiple := mul(1, 29)

Euler power bit3 squared-base reduction floor quotient multiple = 29

Euler power bit3 squared-base reduction normalized remainder := sub(49, 29)

Euler power bit3 squared-base reduction normalized remainder = 20

Euler power bit3 squared-base reduction:ORIGINAL signed integer=49;modulus=29;exact ratio=49/29;floor quotient=1

value=q*n+remainder;normalized residue=20 in[0,29);same congruence class,NOT absolute-value replacement

step=3;previous[accumulator,base,exponent]=[23, 7, 3];low bit=1

next[accumulator,base,exponent]=[16, 20, 1];split exponent=2*q+bit preserves original-power invariant

Euler power bit4 accumulator product := mul(16, 20)

Euler power bit4 accumulator product = 320

Euler power bit4 accumulator reduction exact quotient := div(320, 29)

Euler power bit4 accumulator reduction exact quotient = 320/29

Euler power bit4 accumulator reduction floor quotient multiple := mul(11, 29)

Euler power bit4 accumulator reduction floor quotient multiple = 319

Euler power bit4 accumulator reduction normalized remainder := sub(320, 319)

Euler power bit4 accumulator reduction normalized remainder = 1

Euler power bit4 accumulator reduction:ORIGINAL signed integer=320;modulus=29;exact ratio=320/29;floor quotient=11

value=q*n+remainder;normalized residue=1 in[0,29);same congruence class,NOT absolute-value replacement

Euler power bit4 base square := mul(20, 20)

Euler power bit4 base square = 400

Euler power bit4 squared-base reduction exact quotient := div(400, 29)

Euler power bit4 squared-base reduction exact quotient = 400/29

Euler power bit4 squared-base reduction floor quotient multiple := mul(13, 29)

Euler power bit4 squared-base reduction floor quotient multiple = 377

Euler power bit4 squared-base reduction normalized remainder := sub(400, 377)

Euler power bit4 squared-base reduction normalized remainder = 23

Euler power bit4 squared-base reduction:ORIGINAL signed integer=400;modulus=29;exact ratio=400/29;floor quotient=13

value=q*n+remainder;normalized residue=23 in[0,29);same congruence class,NOT absolute-value replacement

step=4;previous[accumulator,base,exponent]=[16, 20, 1];low bit=1

next[accumulator,base,exponent]=[1, 23, 0];split exponent=2*q+bit preserves original-power invariant

remaining exponent0;final normalized accumulator=1

EXACT original normalized base^28 modulo29=1

Euler ACTUAL base-unit inverse product := mul(3, 10)

Euler ACTUAL base-unit inverse product = 30

Euler ACTUAL inverse check exact quotient := div(30, 29)

Euler ACTUAL inverse check exact quotient = 30/29

Euler ACTUAL inverse check floor quotient multiple := mul(1, 29)

Euler ACTUAL inverse check floor quotient multiple = 29

Euler ACTUAL inverse check normalized remainder := sub(30, 29)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=30;modulus=29;exact ratio=30/29;floor quotient=1

value=q*n+remainder;normalized residue=1 in[0,29);same congruence class,NOT absolute-value replacement

ACTUAL modulus n=29;gcd(original a=-26,n)=1;normalized base=3;ACTUAL inverse=10;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]

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, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] of size phi=28;multiplication by original a=-26 permutes it

multiply ALL unit images:a^(28)*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 modulo29

逐步反向核對

• S091 · 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。
• S090 · 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。
• S089 · 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。
• S088 · 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。
• S087 · 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。
• S086 · 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。
• S085 · 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。
• 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 Euler power bit4 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 Euler power bit4 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 Euler power bit4 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 Euler power bit4 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 Euler power bit4 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 Euler power bit4 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 Euler power bit4 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 Euler power bit4 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 Euler power bit3 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 Euler power bit3 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 Euler power bit3 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 Euler power bit3 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 — 以精確有理數重算 Euler power bit3 accumulator reduction normalized remainder，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 accumulator reduction floor quotient multiple，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 accumulator reduction exact quotient，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Euler power bit3 accumulator product，不以小數近似替代。
• S061 · 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。
• 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 — 以精確有理數重算 Euler power bit2 squared-base reduction normalized remainder，不以小數近似替代。
• S058 · 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。
• S057 · 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。
• S056 · 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。
• 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 — 以精確有理數重算 Euler power bit2 accumulator reduction normalized remainder，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 accumulator reduction floor quotient multiple，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 accumulator reduction exact quotient，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Euler power bit2 accumulator product，不以小數近似替代。
• 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 — 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 — 以精確有理數重算 Euler power bit1 squared-base reduction normalized remainder，不以小數近似替代。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• 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 — 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 — 以精確有理數重算 Euler power bit0 squared-base reduction normalized remainder，不以小數近似替代。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• 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 Euler complete reduced residue count phi，不以小數近似替代。
• 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 — 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 — 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 — 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 — 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 — 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 — 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

(-26, 3, 28, 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。