Question 20100: Multiplication by a Unit Permutes the Complete Reduced Residues

Original question and givens

Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

• G001 — Use integer modulus n=29≥2 and original integer base a=26. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-euler-theorem-number-theory

Step-by-step derivation

source operation=eulerthm-unit-permutation;original snapshot and mathematics retained

explicit ordered source output fields=[units, sorted_mapped_units, permutation, 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(0, 29)

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Euler ORIGINAL signed base normalization normalized remainder := sub(26, 0)

Euler ORIGINAL signed base normalization normalized remainder = 26

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=26;modulus=29;exact ratio=26/29;floor quotient=0

value=q*n+remainder;normalized residue=26 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, No], [1, 1, Yes], [2, 1, Yes], [3, 1, Yes], [4, 1, Yes], [5, 1, Yes], [6, 1, Yes], [7, 1, Yes], [8, 1, Yes], [9, 1, Yes], [10, 1, Yes], [11, 1, Yes], [12, 1, Yes], [13, 1, Yes], [14, 1, Yes], [15, 1, Yes], [16, 1, Yes], [17, 1, Yes], [18, 1, Yes], [19, 1, Yes], [20, 1, Yes], [21, 1, Yes], [22, 1, Yes], [23, 1, Yes], [24, 1, Yes], [25, 1, Yes], [26, 1, Yes], [27, 1, Yes], [28, 1, Yes]]

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

Euler ACTUAL base-unit inverse product := mul(26, 19)

Euler ACTUAL base-unit inverse product = 494

Euler ACTUAL inverse check exact quotient := div(494, 29)

Euler ACTUAL inverse check exact quotient = 494/29

Euler ACTUAL inverse check floor quotient multiple := mul(17, 29)

Euler ACTUAL inverse check floor quotient multiple = 493

Euler ACTUAL inverse check normalized remainder := sub(494, 493)

Euler ACTUAL inverse check normalized remainder = 1

Euler ACTUAL inverse check:ORIGINAL signed integer=494;modulus=29;exact ratio=494/29;floor quotient=17

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=26;ACTUAL inverse=19;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)

Euler ORIGINAL a times unit representative1 := mul(26, 1)

Euler ORIGINAL a times unit representative1 = 26

Euler unit representative1 raw image exact quotient := div(26, 29)

Euler unit representative1 raw image exact quotient = 26/29

Euler unit representative1 raw image floor quotient multiple := mul(0, 29)

Euler unit representative1 raw image floor quotient multiple = 0

Euler unit representative1 raw image normalized remainder := sub(26, 0)

Euler unit representative1 raw image normalized remainder = 26

Euler unit representative1 raw image:ORIGINAL signed integer=26;modulus=29;exact ratio=26/29;floor quotient=0

value=q*n+remainder;normalized residue=26 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative2 := mul(26, 2)

Euler ORIGINAL a times unit representative2 = 52

Euler unit representative2 raw image exact quotient := div(52, 29)

Euler unit representative2 raw image exact quotient = 52/29

Euler unit representative2 raw image floor quotient multiple := mul(1, 29)

Euler unit representative2 raw image floor quotient multiple = 29

Euler unit representative2 raw image normalized remainder := sub(52, 29)

Euler unit representative2 raw image normalized remainder = 23

Euler unit representative2 raw image:ORIGINAL signed integer=52;modulus=29;exact ratio=52/29;floor quotient=1

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

Euler ORIGINAL a times unit representative3 := mul(26, 3)

Euler ORIGINAL a times unit representative3 = 78

Euler unit representative3 raw image exact quotient := div(78, 29)

Euler unit representative3 raw image exact quotient = 78/29

Euler unit representative3 raw image floor quotient multiple := mul(2, 29)

Euler unit representative3 raw image floor quotient multiple = 58

Euler unit representative3 raw image normalized remainder := sub(78, 58)

Euler unit representative3 raw image normalized remainder = 20

Euler unit representative3 raw image:ORIGINAL signed integer=78;modulus=29;exact ratio=78/29;floor quotient=2

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

Euler ORIGINAL a times unit representative4 := mul(26, 4)

Euler ORIGINAL a times unit representative4 = 104

Euler unit representative4 raw image exact quotient := div(104, 29)

Euler unit representative4 raw image exact quotient = 104/29

Euler unit representative4 raw image floor quotient multiple := mul(3, 29)

Euler unit representative4 raw image floor quotient multiple = 87

Euler unit representative4 raw image normalized remainder := sub(104, 87)

Euler unit representative4 raw image normalized remainder = 17

Euler unit representative4 raw image:ORIGINAL signed integer=104;modulus=29;exact ratio=104/29;floor quotient=3

value=q*n+remainder;normalized residue=17 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative5 := mul(26, 5)

Euler ORIGINAL a times unit representative5 = 130

Euler unit representative5 raw image exact quotient := div(130, 29)

Euler unit representative5 raw image exact quotient = 130/29

Euler unit representative5 raw image floor quotient multiple := mul(4, 29)

Euler unit representative5 raw image floor quotient multiple = 116

Euler unit representative5 raw image normalized remainder := sub(130, 116)

Euler unit representative5 raw image normalized remainder = 14

Euler unit representative5 raw image:ORIGINAL signed integer=130;modulus=29;exact ratio=130/29;floor quotient=4

value=q*n+remainder;normalized residue=14 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative6 := mul(26, 6)

Euler ORIGINAL a times unit representative6 = 156

Euler unit representative6 raw image exact quotient := div(156, 29)

Euler unit representative6 raw image exact quotient = 156/29

Euler unit representative6 raw image floor quotient multiple := mul(5, 29)

Euler unit representative6 raw image floor quotient multiple = 145

Euler unit representative6 raw image normalized remainder := sub(156, 145)

Euler unit representative6 raw image normalized remainder = 11

Euler unit representative6 raw image:ORIGINAL signed integer=156;modulus=29;exact ratio=156/29;floor quotient=5

value=q*n+remainder;normalized residue=11 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative7 := mul(26, 7)

Euler ORIGINAL a times unit representative7 = 182

Euler unit representative7 raw image exact quotient := div(182, 29)

Euler unit representative7 raw image exact quotient = 182/29

Euler unit representative7 raw image floor quotient multiple := mul(6, 29)

Euler unit representative7 raw image floor quotient multiple = 174

Euler unit representative7 raw image normalized remainder := sub(182, 174)

Euler unit representative7 raw image normalized remainder = 8

Euler unit representative7 raw image:ORIGINAL signed integer=182;modulus=29;exact ratio=182/29;floor quotient=6

value=q*n+remainder;normalized residue=8 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative8 := mul(26, 8)

Euler ORIGINAL a times unit representative8 = 208

Euler unit representative8 raw image exact quotient := div(208, 29)

Euler unit representative8 raw image exact quotient = 208/29

Euler unit representative8 raw image floor quotient multiple := mul(7, 29)

Euler unit representative8 raw image floor quotient multiple = 203

Euler unit representative8 raw image normalized remainder := sub(208, 203)

Euler unit representative8 raw image normalized remainder = 5

Euler unit representative8 raw image:ORIGINAL signed integer=208;modulus=29;exact ratio=208/29;floor quotient=7

value=q*n+remainder;normalized residue=5 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative9 := mul(26, 9)

Euler ORIGINAL a times unit representative9 = 234

Euler unit representative9 raw image exact quotient := div(234, 29)

Euler unit representative9 raw image exact quotient = 234/29

Euler unit representative9 raw image floor quotient multiple := mul(8, 29)

Euler unit representative9 raw image floor quotient multiple = 232

Euler unit representative9 raw image normalized remainder := sub(234, 232)

Euler unit representative9 raw image normalized remainder = 2

Euler unit representative9 raw image:ORIGINAL signed integer=234;modulus=29;exact ratio=234/29;floor quotient=8

value=q*n+remainder;normalized residue=2 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative10 := mul(26, 10)

Euler ORIGINAL a times unit representative10 = 260

Euler unit representative10 raw image exact quotient := div(260, 29)

Euler unit representative10 raw image exact quotient = 260/29

Euler unit representative10 raw image floor quotient multiple := mul(8, 29)

Euler unit representative10 raw image floor quotient multiple = 232

Euler unit representative10 raw image normalized remainder := sub(260, 232)

Euler unit representative10 raw image normalized remainder = 28

Euler unit representative10 raw image:ORIGINAL signed integer=260;modulus=29;exact ratio=260/29;floor quotient=8

value=q*n+remainder;normalized residue=28 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative11 := mul(26, 11)

Euler ORIGINAL a times unit representative11 = 286

Euler unit representative11 raw image exact quotient := div(286, 29)

Euler unit representative11 raw image exact quotient = 286/29

Euler unit representative11 raw image floor quotient multiple := mul(9, 29)

Euler unit representative11 raw image floor quotient multiple = 261

Euler unit representative11 raw image normalized remainder := sub(286, 261)

Euler unit representative11 raw image normalized remainder = 25

Euler unit representative11 raw image:ORIGINAL signed integer=286;modulus=29;exact ratio=286/29;floor quotient=9

value=q*n+remainder;normalized residue=25 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative12 := mul(26, 12)

Euler ORIGINAL a times unit representative12 = 312

Euler unit representative12 raw image exact quotient := div(312, 29)

Euler unit representative12 raw image exact quotient = 312/29

Euler unit representative12 raw image floor quotient multiple := mul(10, 29)

Euler unit representative12 raw image floor quotient multiple = 290

Euler unit representative12 raw image normalized remainder := sub(312, 290)

Euler unit representative12 raw image normalized remainder = 22

Euler unit representative12 raw image:ORIGINAL signed integer=312;modulus=29;exact ratio=312/29;floor quotient=10

value=q*n+remainder;normalized residue=22 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative13 := mul(26, 13)

Euler ORIGINAL a times unit representative13 = 338

Euler unit representative13 raw image exact quotient := div(338, 29)

Euler unit representative13 raw image exact quotient = 338/29

Euler unit representative13 raw image floor quotient multiple := mul(11, 29)

Euler unit representative13 raw image floor quotient multiple = 319

Euler unit representative13 raw image normalized remainder := sub(338, 319)

Euler unit representative13 raw image normalized remainder = 19

Euler unit representative13 raw image:ORIGINAL signed integer=338;modulus=29;exact ratio=338/29;floor quotient=11

value=q*n+remainder;normalized residue=19 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative14 := mul(26, 14)

Euler ORIGINAL a times unit representative14 = 364

Euler unit representative14 raw image exact quotient := div(364, 29)

Euler unit representative14 raw image exact quotient = 364/29

Euler unit representative14 raw image floor quotient multiple := mul(12, 29)

Euler unit representative14 raw image floor quotient multiple = 348

Euler unit representative14 raw image normalized remainder := sub(364, 348)

Euler unit representative14 raw image normalized remainder = 16

Euler unit representative14 raw image:ORIGINAL signed integer=364;modulus=29;exact ratio=364/29;floor quotient=12

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

Euler ORIGINAL a times unit representative15 := mul(26, 15)

Euler ORIGINAL a times unit representative15 = 390

Euler unit representative15 raw image exact quotient := div(390, 29)

Euler unit representative15 raw image exact quotient = 390/29

Euler unit representative15 raw image floor quotient multiple := mul(13, 29)

Euler unit representative15 raw image floor quotient multiple = 377

Euler unit representative15 raw image normalized remainder := sub(390, 377)

Euler unit representative15 raw image normalized remainder = 13

Euler unit representative15 raw image:ORIGINAL signed integer=390;modulus=29;exact ratio=390/29;floor quotient=13

value=q*n+remainder;normalized residue=13 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative16 := mul(26, 16)

Euler ORIGINAL a times unit representative16 = 416

Euler unit representative16 raw image exact quotient := div(416, 29)

Euler unit representative16 raw image exact quotient = 416/29

Euler unit representative16 raw image floor quotient multiple := mul(14, 29)

Euler unit representative16 raw image floor quotient multiple = 406

Euler unit representative16 raw image normalized remainder := sub(416, 406)

Euler unit representative16 raw image normalized remainder = 10

Euler unit representative16 raw image:ORIGINAL signed integer=416;modulus=29;exact ratio=416/29;floor quotient=14

value=q*n+remainder;normalized residue=10 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative17 := mul(26, 17)

Euler ORIGINAL a times unit representative17 = 442

Euler unit representative17 raw image exact quotient := div(442, 29)

Euler unit representative17 raw image exact quotient = 442/29

Euler unit representative17 raw image floor quotient multiple := mul(15, 29)

Euler unit representative17 raw image floor quotient multiple = 435

Euler unit representative17 raw image normalized remainder := sub(442, 435)

Euler unit representative17 raw image normalized remainder = 7

Euler unit representative17 raw image:ORIGINAL signed integer=442;modulus=29;exact ratio=442/29;floor quotient=15

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

Euler ORIGINAL a times unit representative18 := mul(26, 18)

Euler ORIGINAL a times unit representative18 = 468

Euler unit representative18 raw image exact quotient := div(468, 29)

Euler unit representative18 raw image exact quotient = 468/29

Euler unit representative18 raw image floor quotient multiple := mul(16, 29)

Euler unit representative18 raw image floor quotient multiple = 464

Euler unit representative18 raw image normalized remainder := sub(468, 464)

Euler unit representative18 raw image normalized remainder = 4

Euler unit representative18 raw image:ORIGINAL signed integer=468;modulus=29;exact ratio=468/29;floor quotient=16

value=q*n+remainder;normalized residue=4 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative19 := mul(26, 19)

Euler ORIGINAL a times unit representative19 = 494

Euler unit representative19 raw image exact quotient := div(494, 29)

Euler unit representative19 raw image exact quotient = 494/29

Euler unit representative19 raw image floor quotient multiple := mul(17, 29)

Euler unit representative19 raw image floor quotient multiple = 493

Euler unit representative19 raw image normalized remainder := sub(494, 493)

Euler unit representative19 raw image normalized remainder = 1

Euler unit representative19 raw image:ORIGINAL signed integer=494;modulus=29;exact ratio=494/29;floor quotient=17

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

Euler ORIGINAL a times unit representative20 := mul(26, 20)

Euler ORIGINAL a times unit representative20 = 520

Euler unit representative20 raw image exact quotient := div(520, 29)

Euler unit representative20 raw image exact quotient = 520/29

Euler unit representative20 raw image floor quotient multiple := mul(17, 29)

Euler unit representative20 raw image floor quotient multiple = 493

Euler unit representative20 raw image normalized remainder := sub(520, 493)

Euler unit representative20 raw image normalized remainder = 27

Euler unit representative20 raw image:ORIGINAL signed integer=520;modulus=29;exact ratio=520/29;floor quotient=17

value=q*n+remainder;normalized residue=27 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative21 := mul(26, 21)

Euler ORIGINAL a times unit representative21 = 546

Euler unit representative21 raw image exact quotient := div(546, 29)

Euler unit representative21 raw image exact quotient = 546/29

Euler unit representative21 raw image floor quotient multiple := mul(18, 29)

Euler unit representative21 raw image floor quotient multiple = 522

Euler unit representative21 raw image normalized remainder := sub(546, 522)

Euler unit representative21 raw image normalized remainder = 24

Euler unit representative21 raw image:ORIGINAL signed integer=546;modulus=29;exact ratio=546/29;floor quotient=18

value=q*n+remainder;normalized residue=24 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative22 := mul(26, 22)

Euler ORIGINAL a times unit representative22 = 572

Euler unit representative22 raw image exact quotient := div(572, 29)

Euler unit representative22 raw image exact quotient = 572/29

Euler unit representative22 raw image floor quotient multiple := mul(19, 29)

Euler unit representative22 raw image floor quotient multiple = 551

Euler unit representative22 raw image normalized remainder := sub(572, 551)

Euler unit representative22 raw image normalized remainder = 21

Euler unit representative22 raw image:ORIGINAL signed integer=572;modulus=29;exact ratio=572/29;floor quotient=19

value=q*n+remainder;normalized residue=21 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative23 := mul(26, 23)

Euler ORIGINAL a times unit representative23 = 598

Euler unit representative23 raw image exact quotient := div(598, 29)

Euler unit representative23 raw image exact quotient = 598/29

Euler unit representative23 raw image floor quotient multiple := mul(20, 29)

Euler unit representative23 raw image floor quotient multiple = 580

Euler unit representative23 raw image normalized remainder := sub(598, 580)

Euler unit representative23 raw image normalized remainder = 18

Euler unit representative23 raw image:ORIGINAL signed integer=598;modulus=29;exact ratio=598/29;floor quotient=20

value=q*n+remainder;normalized residue=18 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative24 := mul(26, 24)

Euler ORIGINAL a times unit representative24 = 624

Euler unit representative24 raw image exact quotient := div(624, 29)

Euler unit representative24 raw image exact quotient = 624/29

Euler unit representative24 raw image floor quotient multiple := mul(21, 29)

Euler unit representative24 raw image floor quotient multiple = 609

Euler unit representative24 raw image normalized remainder := sub(624, 609)

Euler unit representative24 raw image normalized remainder = 15

Euler unit representative24 raw image:ORIGINAL signed integer=624;modulus=29;exact ratio=624/29;floor quotient=21

value=q*n+remainder;normalized residue=15 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative25 := mul(26, 25)

Euler ORIGINAL a times unit representative25 = 650

Euler unit representative25 raw image exact quotient := div(650, 29)

Euler unit representative25 raw image exact quotient = 650/29

Euler unit representative25 raw image floor quotient multiple := mul(22, 29)

Euler unit representative25 raw image floor quotient multiple = 638

Euler unit representative25 raw image normalized remainder := sub(650, 638)

Euler unit representative25 raw image normalized remainder = 12

Euler unit representative25 raw image:ORIGINAL signed integer=650;modulus=29;exact ratio=650/29;floor quotient=22

value=q*n+remainder;normalized residue=12 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative26 := mul(26, 26)

Euler ORIGINAL a times unit representative26 = 676

Euler unit representative26 raw image exact quotient := div(676, 29)

Euler unit representative26 raw image exact quotient = 676/29

Euler unit representative26 raw image floor quotient multiple := mul(23, 29)

Euler unit representative26 raw image floor quotient multiple = 667

Euler unit representative26 raw image normalized remainder := sub(676, 667)

Euler unit representative26 raw image normalized remainder = 9

Euler unit representative26 raw image:ORIGINAL signed integer=676;modulus=29;exact ratio=676/29;floor quotient=23

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

Euler ORIGINAL a times unit representative27 := mul(26, 27)

Euler ORIGINAL a times unit representative27 = 702

Euler unit representative27 raw image exact quotient := div(702, 29)

Euler unit representative27 raw image exact quotient = 702/29

Euler unit representative27 raw image floor quotient multiple := mul(24, 29)

Euler unit representative27 raw image floor quotient multiple = 696

Euler unit representative27 raw image normalized remainder := sub(702, 696)

Euler unit representative27 raw image normalized remainder = 6

Euler unit representative27 raw image:ORIGINAL signed integer=702;modulus=29;exact ratio=702/29;floor quotient=24

value=q*n+remainder;normalized residue=6 in[0,29);same congruence class,NOT absolute-value replacement

Euler ORIGINAL a times unit representative28 := mul(26, 28)

Euler ORIGINAL a times unit representative28 = 728

Euler unit representative28 raw image exact quotient := div(728, 29)

Euler unit representative28 raw image exact quotient = 728/29

Euler unit representative28 raw image floor quotient multiple := mul(25, 29)

Euler unit representative28 raw image floor quotient multiple = 725

Euler unit representative28 raw image normalized remainder := sub(728, 725)

Euler unit representative28 raw image normalized remainder = 3

Euler unit representative28 raw image:ORIGINAL signed integer=728;modulus=29;exact ratio=728/29;floor quotient=25

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

COMPLETE original increasing 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];actual images in ORIGINAL unit order=[26, 23, 20, 17, 14, 11, 8, 5, 2, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1, 27, 24, 21, 18, 15, 12, 9, 6, 3]

SOURCE output sorts the mapped list numerically=[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];full-list equality=Yes;not a claim each point is fixed

Reverse checks

• S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S182 · premise_and_result_audit — Recompute Euler unit representative28 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Recompute Euler unit representative28 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S180 · premise_and_result_audit — Recompute Euler unit representative28 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative28 with exact rational arithmetic, not rounded decimals.
• S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S177 · premise_and_result_audit — Recompute Euler unit representative27 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Recompute Euler unit representative27 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S175 · premise_and_result_audit — Recompute Euler unit representative27 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative27 with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S172 · premise_and_result_audit — Recompute Euler unit representative26 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Recompute Euler unit representative26 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Recompute Euler unit representative26 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative26 with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S167 · premise_and_result_audit — Recompute Euler unit representative25 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Recompute Euler unit representative25 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Recompute Euler unit representative25 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative25 with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S162 · premise_and_result_audit — Recompute Euler unit representative24 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Recompute Euler unit representative24 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Recompute Euler unit representative24 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative24 with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S157 · premise_and_result_audit — Recompute Euler unit representative23 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Recompute Euler unit representative23 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Recompute Euler unit representative23 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative23 with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S152 · premise_and_result_audit — Recompute Euler unit representative22 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Recompute Euler unit representative22 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Recompute Euler unit representative22 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative22 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S147 · premise_and_result_audit — Recompute Euler unit representative21 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute Euler unit representative21 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Euler unit representative21 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative21 with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S142 · premise_and_result_audit — Recompute Euler unit representative20 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Euler unit representative20 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Euler unit representative20 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative20 with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S137 · premise_and_result_audit — Recompute Euler unit representative19 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Euler unit representative19 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Euler unit representative19 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative19 with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Recompute Euler unit representative18 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Euler unit representative18 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Euler unit representative18 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative18 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Recompute Euler unit representative17 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Euler unit representative17 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Euler unit representative17 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative17 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Recompute Euler unit representative16 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Euler unit representative16 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Euler unit representative16 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative16 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S117 · premise_and_result_audit — Recompute Euler unit representative15 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Euler unit representative15 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Euler unit representative15 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative15 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S112 · premise_and_result_audit — Recompute Euler unit representative14 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Euler unit representative14 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Euler unit representative14 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative14 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Recompute Euler unit representative13 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Euler unit representative13 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Euler unit representative13 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative13 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Recompute Euler unit representative12 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute Euler unit representative12 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute Euler unit representative12 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative12 with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Recompute Euler unit representative11 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Euler unit representative11 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Euler unit representative11 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative11 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Recompute Euler unit representative10 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Euler unit representative10 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Euler unit representative10 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative10 with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Recompute Euler unit representative9 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Recompute Euler unit representative9 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Euler unit representative9 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative9 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Recompute Euler unit representative8 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Euler unit representative8 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Euler unit representative8 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative8 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Recompute Euler unit representative7 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Euler unit representative7 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Euler unit representative7 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative7 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Recompute Euler unit representative6 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Euler unit representative6 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Euler unit representative6 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative6 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Recompute Euler unit representative5 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Euler unit representative5 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Euler unit representative5 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative5 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Recompute Euler unit representative4 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Euler unit representative4 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Euler unit representative4 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative4 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Recompute Euler unit representative3 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Euler unit representative3 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Euler unit representative3 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative3 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Recompute Euler unit representative2 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Euler unit representative2 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Euler unit representative2 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative2 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Recompute Euler unit representative1 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Euler unit representative1 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Euler unit representative1 raw image exact quotient with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative1 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Euler ACTUAL base-unit inverse product with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S036 · premise_and_result_audit — Recompute Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization normalized remainder with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization floor quotient multiple with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization exact quotient with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([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], [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], Yes, multiplication_by_a_permutes_reduced_residues)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.