第 23397 題：二次剩餘

原題與已知條件

odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={"p":83}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={"p":83}。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-quadratic-residue

逐步推導

source operation=quadratic-square-table;original snapshot and mathematics retained

explicit ordered source output fields=[p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal]

Quadratic ORIGINAL modulus:ORIGINAL integer=83;complete possible trial divisors=[2, 3, 4, 5, 6, 7, 8, 9]

ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 3], [5, 3], [6, 5], [7, 6], [8, 3], [9, 2]];prime=是;a composite>1 has a factor at most its square root

Quadratic ORIGINAL nonzero root1 square := mul(1, 1)

Quadratic ORIGINAL nonzero root1 square = 1

ORIGINAL nonzero root x=1 in1..82;actual square=1

Euclidean square=0*83+1;nonzero square class=1;complete root-index order retained

Quadratic ORIGINAL nonzero root2 square := mul(2, 2)

Quadratic ORIGINAL nonzero root2 square = 4

ORIGINAL nonzero root x=2 in1..82;actual square=4

Euclidean square=0*83+4;nonzero square class=4;complete root-index order retained

Quadratic ORIGINAL nonzero root3 square := mul(3, 3)

Quadratic ORIGINAL nonzero root3 square = 9

ORIGINAL nonzero root x=3 in1..82;actual square=9

Euclidean square=0*83+9;nonzero square class=9;complete root-index order retained

Quadratic ORIGINAL nonzero root4 square := mul(4, 4)

Quadratic ORIGINAL nonzero root4 square = 16

ORIGINAL nonzero root x=4 in1..82;actual square=16

Euclidean square=0*83+16;nonzero square class=16;complete root-index order retained

Quadratic ORIGINAL nonzero root5 square := mul(5, 5)

Quadratic ORIGINAL nonzero root5 square = 25

ORIGINAL nonzero root x=5 in1..82;actual square=25

Euclidean square=0*83+25;nonzero square class=25;complete root-index order retained

Quadratic ORIGINAL nonzero root6 square := mul(6, 6)

Quadratic ORIGINAL nonzero root6 square = 36

ORIGINAL nonzero root x=6 in1..82;actual square=36

Euclidean square=0*83+36;nonzero square class=36;complete root-index order retained

Quadratic ORIGINAL nonzero root7 square := mul(7, 7)

Quadratic ORIGINAL nonzero root7 square = 49

ORIGINAL nonzero root x=7 in1..82;actual square=49

Euclidean square=0*83+49;nonzero square class=49;complete root-index order retained

Quadratic ORIGINAL nonzero root8 square := mul(8, 8)

Quadratic ORIGINAL nonzero root8 square = 64

ORIGINAL nonzero root x=8 in1..82;actual square=64

Euclidean square=0*83+64;nonzero square class=64;complete root-index order retained

Quadratic ORIGINAL nonzero root9 square := mul(9, 9)

Quadratic ORIGINAL nonzero root9 square = 81

ORIGINAL nonzero root x=9 in1..82;actual square=81

Euclidean square=0*83+81;nonzero square class=81;complete root-index order retained

Quadratic ORIGINAL nonzero root10 square := mul(10, 10)

Quadratic ORIGINAL nonzero root10 square = 100

ORIGINAL nonzero root x=10 in1..82;actual square=100

Euclidean square=1*83+17;nonzero square class=17;complete root-index order retained

Quadratic ORIGINAL nonzero root11 square := mul(11, 11)

Quadratic ORIGINAL nonzero root11 square = 121

ORIGINAL nonzero root x=11 in1..82;actual square=121

Euclidean square=1*83+38;nonzero square class=38;complete root-index order retained

Quadratic ORIGINAL nonzero root12 square := mul(12, 12)

Quadratic ORIGINAL nonzero root12 square = 144

ORIGINAL nonzero root x=12 in1..82;actual square=144

Euclidean square=1*83+61;nonzero square class=61;complete root-index order retained

Quadratic ORIGINAL nonzero root13 square := mul(13, 13)

Quadratic ORIGINAL nonzero root13 square = 169

ORIGINAL nonzero root x=13 in1..82;actual square=169

Euclidean square=2*83+3;nonzero square class=3;complete root-index order retained

Quadratic ORIGINAL nonzero root14 square := mul(14, 14)

Quadratic ORIGINAL nonzero root14 square = 196

ORIGINAL nonzero root x=14 in1..82;actual square=196

Euclidean square=2*83+30;nonzero square class=30;complete root-index order retained

Quadratic ORIGINAL nonzero root15 square := mul(15, 15)

Quadratic ORIGINAL nonzero root15 square = 225

ORIGINAL nonzero root x=15 in1..82;actual square=225

Euclidean square=2*83+59;nonzero square class=59;complete root-index order retained

Quadratic ORIGINAL nonzero root16 square := mul(16, 16)

Quadratic ORIGINAL nonzero root16 square = 256

ORIGINAL nonzero root x=16 in1..82;actual square=256

Euclidean square=3*83+7;nonzero square class=7;complete root-index order retained

Quadratic ORIGINAL nonzero root17 square := mul(17, 17)

Quadratic ORIGINAL nonzero root17 square = 289

ORIGINAL nonzero root x=17 in1..82;actual square=289

Euclidean square=3*83+40;nonzero square class=40;complete root-index order retained

Quadratic ORIGINAL nonzero root18 square := mul(18, 18)

Quadratic ORIGINAL nonzero root18 square = 324

ORIGINAL nonzero root x=18 in1..82;actual square=324

Euclidean square=3*83+75;nonzero square class=75;complete root-index order retained

Quadratic ORIGINAL nonzero root19 square := mul(19, 19)

Quadratic ORIGINAL nonzero root19 square = 361

ORIGINAL nonzero root x=19 in1..82;actual square=361

Euclidean square=4*83+29;nonzero square class=29;complete root-index order retained

Quadratic ORIGINAL nonzero root20 square := mul(20, 20)

Quadratic ORIGINAL nonzero root20 square = 400

ORIGINAL nonzero root x=20 in1..82;actual square=400

Euclidean square=4*83+68;nonzero square class=68;complete root-index order retained

Quadratic ORIGINAL nonzero root21 square := mul(21, 21)

Quadratic ORIGINAL nonzero root21 square = 441

ORIGINAL nonzero root x=21 in1..82;actual square=441

Euclidean square=5*83+26;nonzero square class=26;complete root-index order retained

Quadratic ORIGINAL nonzero root22 square := mul(22, 22)

Quadratic ORIGINAL nonzero root22 square = 484

ORIGINAL nonzero root x=22 in1..82;actual square=484

Euclidean square=5*83+69;nonzero square class=69;complete root-index order retained

Quadratic ORIGINAL nonzero root23 square := mul(23, 23)

Quadratic ORIGINAL nonzero root23 square = 529

ORIGINAL nonzero root x=23 in1..82;actual square=529

Euclidean square=6*83+31;nonzero square class=31;complete root-index order retained

Quadratic ORIGINAL nonzero root24 square := mul(24, 24)

Quadratic ORIGINAL nonzero root24 square = 576

ORIGINAL nonzero root x=24 in1..82;actual square=576

Euclidean square=6*83+78;nonzero square class=78;complete root-index order retained

Quadratic ORIGINAL nonzero root25 square := mul(25, 25)

Quadratic ORIGINAL nonzero root25 square = 625

ORIGINAL nonzero root x=25 in1..82;actual square=625

Euclidean square=7*83+44;nonzero square class=44;complete root-index order retained

Quadratic ORIGINAL nonzero root26 square := mul(26, 26)

Quadratic ORIGINAL nonzero root26 square = 676

ORIGINAL nonzero root x=26 in1..82;actual square=676

Euclidean square=8*83+12;nonzero square class=12;complete root-index order retained

Quadratic ORIGINAL nonzero root27 square := mul(27, 27)

Quadratic ORIGINAL nonzero root27 square = 729

ORIGINAL nonzero root x=27 in1..82;actual square=729

Euclidean square=8*83+65;nonzero square class=65;complete root-index order retained

Quadratic ORIGINAL nonzero root28 square := mul(28, 28)

Quadratic ORIGINAL nonzero root28 square = 784

ORIGINAL nonzero root x=28 in1..82;actual square=784

Euclidean square=9*83+37;nonzero square class=37;complete root-index order retained

Quadratic ORIGINAL nonzero root29 square := mul(29, 29)

Quadratic ORIGINAL nonzero root29 square = 841

ORIGINAL nonzero root x=29 in1..82;actual square=841

Euclidean square=10*83+11;nonzero square class=11;complete root-index order retained

Quadratic ORIGINAL nonzero root30 square := mul(30, 30)

Quadratic ORIGINAL nonzero root30 square = 900

ORIGINAL nonzero root x=30 in1..82;actual square=900

Euclidean square=10*83+70;nonzero square class=70;complete root-index order retained

Quadratic ORIGINAL nonzero root31 square := mul(31, 31)

Quadratic ORIGINAL nonzero root31 square = 961

ORIGINAL nonzero root x=31 in1..82;actual square=961

Euclidean square=11*83+48;nonzero square class=48;complete root-index order retained

Quadratic ORIGINAL nonzero root32 square := mul(32, 32)

Quadratic ORIGINAL nonzero root32 square = 1024

ORIGINAL nonzero root x=32 in1..82;actual square=1024

Euclidean square=12*83+28;nonzero square class=28;complete root-index order retained

Quadratic ORIGINAL nonzero root33 square := mul(33, 33)

Quadratic ORIGINAL nonzero root33 square = 1089

ORIGINAL nonzero root x=33 in1..82;actual square=1089

Euclidean square=13*83+10;nonzero square class=10;complete root-index order retained

Quadratic ORIGINAL nonzero root34 square := mul(34, 34)

Quadratic ORIGINAL nonzero root34 square = 1156

ORIGINAL nonzero root x=34 in1..82;actual square=1156

Euclidean square=13*83+77;nonzero square class=77;complete root-index order retained

Quadratic ORIGINAL nonzero root35 square := mul(35, 35)

Quadratic ORIGINAL nonzero root35 square = 1225

ORIGINAL nonzero root x=35 in1..82;actual square=1225

Euclidean square=14*83+63;nonzero square class=63;complete root-index order retained

Quadratic ORIGINAL nonzero root36 square := mul(36, 36)

Quadratic ORIGINAL nonzero root36 square = 1296

ORIGINAL nonzero root x=36 in1..82;actual square=1296

Euclidean square=15*83+51;nonzero square class=51;complete root-index order retained

Quadratic ORIGINAL nonzero root37 square := mul(37, 37)

Quadratic ORIGINAL nonzero root37 square = 1369

ORIGINAL nonzero root x=37 in1..82;actual square=1369

Euclidean square=16*83+41;nonzero square class=41;complete root-index order retained

Quadratic ORIGINAL nonzero root38 square := mul(38, 38)

Quadratic ORIGINAL nonzero root38 square = 1444

ORIGINAL nonzero root x=38 in1..82;actual square=1444

Euclidean square=17*83+33;nonzero square class=33;complete root-index order retained

Quadratic ORIGINAL nonzero root39 square := mul(39, 39)

Quadratic ORIGINAL nonzero root39 square = 1521

ORIGINAL nonzero root x=39 in1..82;actual square=1521

Euclidean square=18*83+27;nonzero square class=27;complete root-index order retained

Quadratic ORIGINAL nonzero root40 square := mul(40, 40)

Quadratic ORIGINAL nonzero root40 square = 1600

ORIGINAL nonzero root x=40 in1..82;actual square=1600

Euclidean square=19*83+23;nonzero square class=23;complete root-index order retained

Quadratic ORIGINAL nonzero root41 square := mul(41, 41)

Quadratic ORIGINAL nonzero root41 square = 1681

ORIGINAL nonzero root x=41 in1..82;actual square=1681

Euclidean square=20*83+21;nonzero square class=21;complete root-index order retained

Quadratic ORIGINAL nonzero root42 square := mul(42, 42)

Quadratic ORIGINAL nonzero root42 square = 1764

ORIGINAL nonzero root x=42 in1..82;actual square=1764

Euclidean square=21*83+21;nonzero square class=21;complete root-index order retained

Quadratic ORIGINAL nonzero root43 square := mul(43, 43)

Quadratic ORIGINAL nonzero root43 square = 1849

ORIGINAL nonzero root x=43 in1..82;actual square=1849

Euclidean square=22*83+23;nonzero square class=23;complete root-index order retained

Quadratic ORIGINAL nonzero root44 square := mul(44, 44)

Quadratic ORIGINAL nonzero root44 square = 1936

ORIGINAL nonzero root x=44 in1..82;actual square=1936

Euclidean square=23*83+27;nonzero square class=27;complete root-index order retained

Quadratic ORIGINAL nonzero root45 square := mul(45, 45)

Quadratic ORIGINAL nonzero root45 square = 2025

ORIGINAL nonzero root x=45 in1..82;actual square=2025

Euclidean square=24*83+33;nonzero square class=33;complete root-index order retained

Quadratic ORIGINAL nonzero root46 square := mul(46, 46)

Quadratic ORIGINAL nonzero root46 square = 2116

ORIGINAL nonzero root x=46 in1..82;actual square=2116

Euclidean square=25*83+41;nonzero square class=41;complete root-index order retained

Quadratic ORIGINAL nonzero root47 square := mul(47, 47)

Quadratic ORIGINAL nonzero root47 square = 2209

ORIGINAL nonzero root x=47 in1..82;actual square=2209

Euclidean square=26*83+51;nonzero square class=51;complete root-index order retained

Quadratic ORIGINAL nonzero root48 square := mul(48, 48)

Quadratic ORIGINAL nonzero root48 square = 2304

ORIGINAL nonzero root x=48 in1..82;actual square=2304

Euclidean square=27*83+63;nonzero square class=63;complete root-index order retained

Quadratic ORIGINAL nonzero root49 square := mul(49, 49)

Quadratic ORIGINAL nonzero root49 square = 2401

ORIGINAL nonzero root x=49 in1..82;actual square=2401

Euclidean square=28*83+77;nonzero square class=77;complete root-index order retained

Quadratic ORIGINAL nonzero root50 square := mul(50, 50)

Quadratic ORIGINAL nonzero root50 square = 2500

ORIGINAL nonzero root x=50 in1..82;actual square=2500

Euclidean square=30*83+10;nonzero square class=10;complete root-index order retained

Quadratic ORIGINAL nonzero root51 square := mul(51, 51)

Quadratic ORIGINAL nonzero root51 square = 2601

ORIGINAL nonzero root x=51 in1..82;actual square=2601

Euclidean square=31*83+28;nonzero square class=28;complete root-index order retained

Quadratic ORIGINAL nonzero root52 square := mul(52, 52)

Quadratic ORIGINAL nonzero root52 square = 2704

ORIGINAL nonzero root x=52 in1..82;actual square=2704

Euclidean square=32*83+48;nonzero square class=48;complete root-index order retained

Quadratic ORIGINAL nonzero root53 square := mul(53, 53)

Quadratic ORIGINAL nonzero root53 square = 2809

ORIGINAL nonzero root x=53 in1..82;actual square=2809

Euclidean square=33*83+70;nonzero square class=70;complete root-index order retained

Quadratic ORIGINAL nonzero root54 square := mul(54, 54)

Quadratic ORIGINAL nonzero root54 square = 2916

ORIGINAL nonzero root x=54 in1..82;actual square=2916

Euclidean square=35*83+11;nonzero square class=11;complete root-index order retained

Quadratic ORIGINAL nonzero root55 square := mul(55, 55)

Quadratic ORIGINAL nonzero root55 square = 3025

ORIGINAL nonzero root x=55 in1..82;actual square=3025

Euclidean square=36*83+37;nonzero square class=37;complete root-index order retained

Quadratic ORIGINAL nonzero root56 square := mul(56, 56)

Quadratic ORIGINAL nonzero root56 square = 3136

ORIGINAL nonzero root x=56 in1..82;actual square=3136

Euclidean square=37*83+65;nonzero square class=65;complete root-index order retained

Quadratic ORIGINAL nonzero root57 square := mul(57, 57)

Quadratic ORIGINAL nonzero root57 square = 3249

ORIGINAL nonzero root x=57 in1..82;actual square=3249

Euclidean square=39*83+12;nonzero square class=12;complete root-index order retained

Quadratic ORIGINAL nonzero root58 square := mul(58, 58)

Quadratic ORIGINAL nonzero root58 square = 3364

ORIGINAL nonzero root x=58 in1..82;actual square=3364

Euclidean square=40*83+44;nonzero square class=44;complete root-index order retained

Quadratic ORIGINAL nonzero root59 square := mul(59, 59)

Quadratic ORIGINAL nonzero root59 square = 3481

ORIGINAL nonzero root x=59 in1..82;actual square=3481

Euclidean square=41*83+78;nonzero square class=78;complete root-index order retained

Quadratic ORIGINAL nonzero root60 square := mul(60, 60)

Quadratic ORIGINAL nonzero root60 square = 3600

ORIGINAL nonzero root x=60 in1..82;actual square=3600

Euclidean square=43*83+31;nonzero square class=31;complete root-index order retained

Quadratic ORIGINAL nonzero root61 square := mul(61, 61)

Quadratic ORIGINAL nonzero root61 square = 3721

ORIGINAL nonzero root x=61 in1..82;actual square=3721

Euclidean square=44*83+69;nonzero square class=69;complete root-index order retained

Quadratic ORIGINAL nonzero root62 square := mul(62, 62)

Quadratic ORIGINAL nonzero root62 square = 3844

ORIGINAL nonzero root x=62 in1..82;actual square=3844

Euclidean square=46*83+26;nonzero square class=26;complete root-index order retained

Quadratic ORIGINAL nonzero root63 square := mul(63, 63)

Quadratic ORIGINAL nonzero root63 square = 3969

ORIGINAL nonzero root x=63 in1..82;actual square=3969

Euclidean square=47*83+68;nonzero square class=68;complete root-index order retained

Quadratic ORIGINAL nonzero root64 square := mul(64, 64)

Quadratic ORIGINAL nonzero root64 square = 4096

ORIGINAL nonzero root x=64 in1..82;actual square=4096

Euclidean square=49*83+29;nonzero square class=29;complete root-index order retained

Quadratic ORIGINAL nonzero root65 square := mul(65, 65)

Quadratic ORIGINAL nonzero root65 square = 4225

ORIGINAL nonzero root x=65 in1..82;actual square=4225

Euclidean square=50*83+75;nonzero square class=75;complete root-index order retained

Quadratic ORIGINAL nonzero root66 square := mul(66, 66)

Quadratic ORIGINAL nonzero root66 square = 4356

ORIGINAL nonzero root x=66 in1..82;actual square=4356

Euclidean square=52*83+40;nonzero square class=40;complete root-index order retained

Quadratic ORIGINAL nonzero root67 square := mul(67, 67)

Quadratic ORIGINAL nonzero root67 square = 4489

ORIGINAL nonzero root x=67 in1..82;actual square=4489

Euclidean square=54*83+7;nonzero square class=7;complete root-index order retained

Quadratic ORIGINAL nonzero root68 square := mul(68, 68)

Quadratic ORIGINAL nonzero root68 square = 4624

ORIGINAL nonzero root x=68 in1..82;actual square=4624

Euclidean square=55*83+59;nonzero square class=59;complete root-index order retained

Quadratic ORIGINAL nonzero root69 square := mul(69, 69)

Quadratic ORIGINAL nonzero root69 square = 4761

ORIGINAL nonzero root x=69 in1..82;actual square=4761

Euclidean square=57*83+30;nonzero square class=30;complete root-index order retained

Quadratic ORIGINAL nonzero root70 square := mul(70, 70)

Quadratic ORIGINAL nonzero root70 square = 4900

ORIGINAL nonzero root x=70 in1..82;actual square=4900

Euclidean square=59*83+3;nonzero square class=3;complete root-index order retained

Quadratic ORIGINAL nonzero root71 square := mul(71, 71)

Quadratic ORIGINAL nonzero root71 square = 5041

ORIGINAL nonzero root x=71 in1..82;actual square=5041

Euclidean square=60*83+61;nonzero square class=61;complete root-index order retained

Quadratic ORIGINAL nonzero root72 square := mul(72, 72)

Quadratic ORIGINAL nonzero root72 square = 5184

ORIGINAL nonzero root x=72 in1..82;actual square=5184

Euclidean square=62*83+38;nonzero square class=38;complete root-index order retained

Quadratic ORIGINAL nonzero root73 square := mul(73, 73)

Quadratic ORIGINAL nonzero root73 square = 5329

ORIGINAL nonzero root x=73 in1..82;actual square=5329

Euclidean square=64*83+17;nonzero square class=17;complete root-index order retained

Quadratic ORIGINAL nonzero root74 square := mul(74, 74)

Quadratic ORIGINAL nonzero root74 square = 5476

ORIGINAL nonzero root x=74 in1..82;actual square=5476

Euclidean square=65*83+81;nonzero square class=81;complete root-index order retained

Quadratic ORIGINAL nonzero root75 square := mul(75, 75)

Quadratic ORIGINAL nonzero root75 square = 5625

ORIGINAL nonzero root x=75 in1..82;actual square=5625

Euclidean square=67*83+64;nonzero square class=64;complete root-index order retained

Quadratic ORIGINAL nonzero root76 square := mul(76, 76)

Quadratic ORIGINAL nonzero root76 square = 5776

ORIGINAL nonzero root x=76 in1..82;actual square=5776

Euclidean square=69*83+49;nonzero square class=49;complete root-index order retained

Quadratic ORIGINAL nonzero root77 square := mul(77, 77)

Quadratic ORIGINAL nonzero root77 square = 5929

ORIGINAL nonzero root x=77 in1..82;actual square=5929

Euclidean square=71*83+36;nonzero square class=36;complete root-index order retained

Quadratic ORIGINAL nonzero root78 square := mul(78, 78)

Quadratic ORIGINAL nonzero root78 square = 6084

ORIGINAL nonzero root x=78 in1..82;actual square=6084

Euclidean square=73*83+25;nonzero square class=25;complete root-index order retained

Quadratic ORIGINAL nonzero root79 square := mul(79, 79)

Quadratic ORIGINAL nonzero root79 square = 6241

ORIGINAL nonzero root x=79 in1..82;actual square=6241

Euclidean square=75*83+16;nonzero square class=16;complete root-index order retained

Quadratic ORIGINAL nonzero root80 square := mul(80, 80)

Quadratic ORIGINAL nonzero root80 square = 6400

ORIGINAL nonzero root x=80 in1..82;actual square=6400

Euclidean square=77*83+9;nonzero square class=9;complete root-index order retained

Quadratic ORIGINAL nonzero root81 square := mul(81, 81)

Quadratic ORIGINAL nonzero root81 square = 6561

ORIGINAL nonzero root x=81 in1..82;actual square=6561

Euclidean square=79*83+4;nonzero square class=4;complete root-index order retained

Quadratic ORIGINAL nonzero root82 square := mul(82, 82)

Quadratic ORIGINAL nonzero root82 square = 6724

ORIGINAL nonzero root x=82 in1..82;actual square=6724

Euclidean square=81*83+1;nonzero square class=1;complete root-index order retained

Quadratic distinct nonzero residue count := 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Quadratic distinct nonzero residue count = 41

Quadratic ORIGINAL modulus minus1 := sub(83, 1)

Quadratic ORIGINAL modulus minus1 = 82

Quadratic half of all nonzero roots := div(82, 2)

Quadratic half of all nonzero roots = 41

ORIGINAL ODD PRIME p=83;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 17], [11, 38], [12, 61], [13, 3], [14, 30], [15, 59], [16, 7], [17, 40], [18, 75], [19, 29], [20, 68], [21, 26], [22, 69], [23, 31], [24, 78], [25, 44], [26, 12], [27, 65], [28, 37], [29, 11], [30, 70], [31, 48], [32, 28], [33, 10], [34, 77], [35, 63], [36, 51], [37, 41], [38, 33], [39, 27], [40, 23], [41, 21], [42, 21], [43, 23], [44, 27], [45, 33], [46, 41], [47, 51], [48, 63], [49, 77], [50, 10], [51, 28], [52, 48], [53, 70], [54, 11], [55, 37], [56, 65], [57, 12], [58, 44], [59, 78], [60, 31], [61, 69], [62, 26], [63, 68], [64, 29], [65, 75], [66, 40], [67, 7], [68, 59], [69, 30], [70, 3], [71, 61], [72, 38], [73, 17], [74, 81], [75, 64], [76, 49], [77, 36], [78, 25], [79, 16], [80, 9], [81, 4], [82, 1]];distinct classes=[1, 3, 4, 7, 9, 10, 11, 12, 16, 17, 21, 23, 25, 26, 27, 28, 29, 30, 31, 33, 36, 37, 38, 40, 41, 44, 48, 49, 51, 59, 61, 63, 64, 65, 68, 69, 70, 75, 77, 78, 81]

x^2=y^2 implies p divides(x-y)(x+y),so x=y or x=-y modp;odd p and nonzero x make opposite roots DISTINCT;each image has2 roots,so residue count=41=(83-1)/2=41

逐步反向核對

• S170 · 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。
• S169 · premise_and_result_audit — 以精確有理數重算 Quadratic half of all nonzero roots，不以小數近似替代。
• S168 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL modulus minus1，不以小數近似替代。
• S167 · premise_and_result_audit — 以精確有理數重算 Quadratic distinct nonzero residue count，不以小數近似替代。
• S166 · 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。
• S165 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root82 square，不以小數近似替代。
• S164 · 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。
• S163 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root81 square，不以小數近似替代。
• S162 · 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。
• S161 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root80 square，不以小數近似替代。
• S160 · 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。
• S159 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root79 square，不以小數近似替代。
• S158 · 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。
• S157 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root78 square，不以小數近似替代。
• S156 · 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。
• S155 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root77 square，不以小數近似替代。
• S154 · 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。
• S153 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root76 square，不以小數近似替代。
• S152 · 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。
• S151 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root75 square，不以小數近似替代。
• S150 · 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。
• S149 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root74 square，不以小數近似替代。
• S148 · 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。
• S147 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root73 square，不以小數近似替代。
• S146 · 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。
• S145 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root72 square，不以小數近似替代。
• S144 · 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。
• S143 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root71 square，不以小數近似替代。
• S142 · 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。
• S141 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root70 square，不以小數近似替代。
• S140 · 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。
• S139 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root69 square，不以小數近似替代。
• S138 · 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。
• S137 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root68 square，不以小數近似替代。
• S136 · 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。
• S135 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root67 square，不以小數近似替代。
• S134 · 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。
• S133 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root66 square，不以小數近似替代。
• S132 · 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。
• S131 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root65 square，不以小數近似替代。
• S130 · 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。
• S129 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root64 square，不以小數近似替代。
• S128 · 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。
• S127 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root63 square，不以小數近似替代。
• S126 · 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。
• S125 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root62 square，不以小數近似替代。
• S124 · 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。
• S123 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root61 square，不以小數近似替代。
• S122 · 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。
• S121 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root60 square，不以小數近似替代。
• S120 · 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。
• S119 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root59 square，不以小數近似替代。
• S118 · 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。
• S117 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root58 square，不以小數近似替代。
• S116 · 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。
• S115 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root57 square，不以小數近似替代。
• S114 · 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。
• S113 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root56 square，不以小數近似替代。
• S112 · 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。
• S111 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root55 square，不以小數近似替代。
• S110 · 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。
• S109 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root54 square，不以小數近似替代。
• S108 · 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。
• S107 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root53 square，不以小數近似替代。
• S106 · 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。
• S105 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root52 square，不以小數近似替代。
• S104 · 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。
• S103 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root51 square，不以小數近似替代。
• S102 · 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。
• S101 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root50 square，不以小數近似替代。
• S100 · 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。
• S099 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root49 square，不以小數近似替代。
• S098 · 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。
• S097 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root48 square，不以小數近似替代。
• S096 · 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。
• S095 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root47 square，不以小數近似替代。
• S094 · 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。
• S093 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root46 square，不以小數近似替代。
• S092 · 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。
• S091 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root45 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root44 square，不以小數近似替代。
• S088 · 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。
• S087 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root43 square，不以小數近似替代。
• S086 · 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。
• S085 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root42 square，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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root41 square，不以小數近似替代。
• 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 Quadratic ORIGINAL nonzero root40 square，不以小數近似替代。
• S080 · 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。
• S079 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root39 square，不以小數近似替代。
• S078 · 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。
• S077 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root38 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root37 square，不以小數近似替代。
• 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 Quadratic ORIGINAL nonzero root36 square，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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root35 square，不以小數近似替代。
• S070 · 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。
• S069 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root34 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root33 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 Quadratic ORIGINAL nonzero root32 square，不以小數近似替代。
• S064 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S063 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root31 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 Quadratic ORIGINAL nonzero root30 square，不以小數近似替代。
• S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S059 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root29 square，不以小數近似替代。
• S058 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S057 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root28 square，不以小數近似替代。
• 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 Quadratic ORIGINAL nonzero root27 square，不以小數近似替代。
• S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S053 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root26 square，不以小數近似替代。
• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root25 square，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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root24 square，不以小數近似替代。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root23 square，不以小數近似替代。
• S046 · 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。
• S045 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root22 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root21 square，不以小數近似替代。
• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root20 square，不以小數近似替代。
• S040 · 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。
• S039 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root19 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root18 square，不以小數近似替代。
• S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S035 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root17 square，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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root16 square，不以小數近似替代。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root15 square，不以小數近似替代。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root14 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root13 square，不以小數近似替代。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root12 square，不以小數近似替代。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root11 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root10 square，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root9 square，不以小數近似替代。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root8 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root7 square，不以小數近似替代。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root6 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root5 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root4 square，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root3 square，不以小數近似替代。
• 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 — 以精確有理數重算 Quadratic ORIGINAL nonzero root2 square，不以小數近似替代。
• S004 · 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。
• S003 · premise_and_result_audit — 以精確有理數重算 Quadratic ORIGINAL nonzero root1 square，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(83, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 17], [11, 38], [12, 61], [13, 3], [14, 30], [15, 59], [16, 7], [17, 40], [18, 75], [19, 29], [20, 68], [21, 26], [22, 69], [23, 31], [24, 78], [25, 44], [26, 12], [27, 65], [28, 37], [29, 11], [30, 70], [31, 48], [32, 28], [33, 10], [34, 77], [35, 63], [36, 51], [37, 41], [38, 33], [39, 27], [40, 23], [41, 21], [42, 21], [43, 23], [44, 27], [45, 33], [46, 41], [47, 51], [48, 63], [49, 77], [50, 10], [51, 28], [52, 48], [53, 70], [54, 11], [55, 37], [56, 65], [57, 12], [58, 44], [59, 78], [60, 31], [61, 69], [62, 26], [63, 68], [64, 29], [65, 75], [66, 40], [67, 7], [68, 59], [69, 30], [70, 3], [71, 61], [72, 38], [73, 17], [74, 81], [75, 64], [76, 49], [77, 36], [78, 25], [79, 16], [80, 9], [81, 4], [82, 1]], [1, 3, 4, 7, 9, 10, 11, 12, 16, 17, 21, 23, 25, 26, 27, 28, 29, 30, 31, 33, 36, 37, 38, 40, 41, 44, 48, 49, 51, 59, 61, 63, 64, 65, 68, 69, 70, 75, 77, 78, 81], 41, 41, 是)

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。