Question 24192: Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Original question and givens

Use odd prime p=61, original a=45, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

• G001 — Use odd prime p=61, original a=45, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, 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-criterion-quadratic-residue

Step-by-step derivation

source roles = [eulercrit-residue-nonresidue-cases]

derived = [[prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate]]

source roles = [61]

derived = [[[2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 5]], Yes]

Euler ORIGINAL signed quotient times modulus := mul(0, 61)

Euler ORIGINAL signed quotient times modulus = 0

Euler ORIGINAL signed a minus modulus multiple := sub(45, 0)

Euler ORIGINAL signed a minus modulus multiple = 45

source roles = [45, 61, 0]

derived = [45, Yes]

source roles = [61, 45, 45]

derived = [Yes, Yes]

Euler square root candidate1 := mul(1, 1)

Euler square root candidate1 = 1

Euler square root candidate2 := mul(2, 2)

Euler square root candidate2 = 4

Euler square root candidate3 := mul(3, 3)

Euler square root candidate3 = 9

Euler square root candidate4 := mul(4, 4)

Euler square root candidate4 = 16

Euler square root candidate5 := mul(5, 5)

Euler square root candidate5 = 25

Euler square root candidate6 := mul(6, 6)

Euler square root candidate6 = 36

Euler square root candidate7 := mul(7, 7)

Euler square root candidate7 = 49

Euler square root candidate8 := mul(8, 8)

Euler square root candidate8 = 64

Euler square root candidate9 := mul(9, 9)

Euler square root candidate9 = 81

Euler square root candidate10 := mul(10, 10)

Euler square root candidate10 = 100

Euler square root candidate11 := mul(11, 11)

Euler square root candidate11 = 121

Euler square root candidate12 := mul(12, 12)

Euler square root candidate12 = 144

Euler square root candidate13 := mul(13, 13)

Euler square root candidate13 = 169

Euler square root candidate14 := mul(14, 14)

Euler square root candidate14 = 196

Euler square root candidate15 := mul(15, 15)

Euler square root candidate15 = 225

Euler square root candidate16 := mul(16, 16)

Euler square root candidate16 = 256

Euler square root candidate17 := mul(17, 17)

Euler square root candidate17 = 289

Euler square root candidate18 := mul(18, 18)

Euler square root candidate18 = 324

Euler square root candidate19 := mul(19, 19)

Euler square root candidate19 = 361

Euler square root candidate20 := mul(20, 20)

Euler square root candidate20 = 400

Euler square root candidate21 := mul(21, 21)

Euler square root candidate21 = 441

Euler square root candidate22 := mul(22, 22)

Euler square root candidate22 = 484

Euler square root candidate23 := mul(23, 23)

Euler square root candidate23 = 529

Euler square root candidate24 := mul(24, 24)

Euler square root candidate24 = 576

Euler square root candidate25 := mul(25, 25)

Euler square root candidate25 = 625

Euler square root candidate26 := mul(26, 26)

Euler square root candidate26 = 676

Euler square root candidate27 := mul(27, 27)

Euler square root candidate27 = 729

Euler square root candidate28 := mul(28, 28)

Euler square root candidate28 = 784

Euler square root candidate29 := mul(29, 29)

Euler square root candidate29 = 841

Euler square root candidate30 := mul(30, 30)

Euler square root candidate30 = 900

Euler square root candidate31 := mul(31, 31)

Euler square root candidate31 = 961

Euler square root candidate32 := mul(32, 32)

Euler square root candidate32 = 1024

Euler square root candidate33 := mul(33, 33)

Euler square root candidate33 = 1089

Euler square root candidate34 := mul(34, 34)

Euler square root candidate34 = 1156

Euler square root candidate35 := mul(35, 35)

Euler square root candidate35 = 1225

Euler square root candidate36 := mul(36, 36)

Euler square root candidate36 = 1296

Euler square root candidate37 := mul(37, 37)

Euler square root candidate37 = 1369

Euler square root candidate38 := mul(38, 38)

Euler square root candidate38 = 1444

Euler square root candidate39 := mul(39, 39)

Euler square root candidate39 = 1521

Euler square root candidate40 := mul(40, 40)

Euler square root candidate40 = 1600

Euler square root candidate41 := mul(41, 41)

Euler square root candidate41 = 1681

Euler square root candidate42 := mul(42, 42)

Euler square root candidate42 = 1764

Euler square root candidate43 := mul(43, 43)

Euler square root candidate43 = 1849

Euler square root candidate44 := mul(44, 44)

Euler square root candidate44 = 1936

Euler square root candidate45 := mul(45, 45)

Euler square root candidate45 = 2025

Euler square root candidate46 := mul(46, 46)

Euler square root candidate46 = 2116

Euler square root candidate47 := mul(47, 47)

Euler square root candidate47 = 2209

Euler square root candidate48 := mul(48, 48)

Euler square root candidate48 = 2304

Euler square root candidate49 := mul(49, 49)

Euler square root candidate49 = 2401

Euler square root candidate50 := mul(50, 50)

Euler square root candidate50 = 2500

Euler square root candidate51 := mul(51, 51)

Euler square root candidate51 = 2601

Euler square root candidate52 := mul(52, 52)

Euler square root candidate52 = 2704

Euler square root candidate53 := mul(53, 53)

Euler square root candidate53 = 2809

Euler square root candidate54 := mul(54, 54)

Euler square root candidate54 = 2916

Euler square root candidate55 := mul(55, 55)

Euler square root candidate55 = 3025

Euler square root candidate56 := mul(56, 56)

Euler square root candidate56 = 3136

Euler square root candidate57 := mul(57, 57)

Euler square root candidate57 = 3249

Euler square root candidate58 := mul(58, 58)

Euler square root candidate58 = 3364

Euler square root candidate59 := mul(59, 59)

Euler square root candidate59 = 3481

Euler square root candidate60 := mul(60, 60)

Euler square root candidate60 = 3600

source roles = [61, 45]

derived = [[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 3], [9, 20], [10, 39], [11, 60], [12, 22], [13, 47], [14, 13], [15, 42], [16, 12], [17, 45], [18, 19], [19, 56], [20, 34], [21, 14], [22, 57], [23, 41], [24, 27], [25, 15], [26, 5], [27, 58], [28, 52], [29, 48], [30, 46], [31, 46], [32, 48], [33, 52], [34, 58], [35, 5], [36, 15], [37, 27], [38, 41], [39, 57], [40, 14], [41, 34], [42, 56], [43, 19], [44, 45], [45, 12], [46, 42], [47, 13], [48, 47], [49, 22], [50, 60], [51, 39], [52, 20], [53, 3], [54, 49], [55, 36], [56, 25], [57, 16], [58, 9], [59, 4], [60, 1]], [17, 44], 1]

Euler binary power square0 := mul(45, 45)

Euler binary power square0 = 2025

source roles = [0, [1, 45, 30], 61, 45, 30]

derived = [null, 2025, 1, 12, 15]

Euler binary power accumulate1 := mul(1, 12)

Euler binary power accumulate1 = 12

Euler binary power square1 := mul(12, 12)

Euler binary power square1 = 144

source roles = [1, [1, 12, 15], 61, 45, 30]

derived = [12, 144, 12, 22, 7]

Euler binary power accumulate2 := mul(12, 22)

Euler binary power accumulate2 = 264

Euler binary power square2 := mul(22, 22)

Euler binary power square2 = 484

source roles = [2, [12, 22, 7], 61, 45, 30]

derived = [264, 484, 20, 57, 3]

Euler binary power accumulate3 := mul(20, 57)

Euler binary power accumulate3 = 1140

Euler binary power square3 := mul(57, 57)

Euler binary power square3 = 3249

source roles = [3, [20, 57, 3], 61, 45, 30]

derived = [1140, 3249, 42, 16, 1]

Euler binary power accumulate4 := mul(42, 16)

Euler binary power accumulate4 = 672

Euler binary power square4 := mul(16, 16)

Euler binary power square4 = 256

source roles = [4, [42, 16, 1], 61, 45, 30]

derived = [672, 256, 1, 12, 0]

source roles = [61, [17, 44], 1, 1]

derived = [1, Yes, one finite instance is not a general theorem proof]

Reverse checks

• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Recompute Euler binary power square4 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Euler binary power accumulate4 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 binary power square3 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Euler binary power accumulate3 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Recompute Euler binary power square2 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Euler binary power accumulate2 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Recompute Euler binary power square1 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Euler binary power accumulate1 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Recompute Euler binary power square0 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Recompute Euler square root candidate60 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Euler square root candidate59 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Euler square root candidate58 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Euler square root candidate57 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Euler square root candidate56 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Euler square root candidate55 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Euler square root candidate54 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Euler square root candidate53 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Euler square root candidate52 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Euler square root candidate51 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Euler square root candidate50 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Euler square root candidate49 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Euler square root candidate48 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Euler square root candidate47 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Euler square root candidate46 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Euler square root candidate45 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Euler square root candidate44 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Euler square root candidate43 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Euler square root candidate42 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Euler square root candidate41 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Euler square root candidate40 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Euler square root candidate39 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Euler square root candidate38 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Euler square root candidate37 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Euler square root candidate36 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Euler square root candidate35 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Euler square root candidate34 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Euler square root candidate33 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Euler square root candidate32 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Euler square root candidate31 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Euler square root candidate30 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Euler square root candidate29 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Euler square root candidate28 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Euler square root candidate27 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Euler square root candidate26 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Euler square root candidate25 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Euler square root candidate24 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Euler square root candidate23 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Euler square root candidate22 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Euler square root candidate21 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Euler square root candidate20 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Euler square root candidate19 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Euler square root candidate18 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Euler square root candidate17 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Euler square root candidate16 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Euler square root candidate15 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Euler square root candidate14 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Euler square root candidate13 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Euler square root candidate12 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Euler square root candidate11 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Euler square root candidate10 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Euler square root candidate9 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Euler square root candidate8 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Euler square root candidate7 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Euler square root candidate6 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Euler square root candidate5 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Euler square root candidate4 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Euler square root candidate3 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Euler square root candidate2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Euler square root candidate1 with exact rational arithmetic, not rounded decimals.
• 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 a minus modulus multiple with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed quotient times modulus with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(61, 45, quadratic_residue, [17, 44], 1, 1, Yes, residue_maps_to_plus_one)

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.