Question 24196: Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Original question and givens

Use odd prime p=79, original a=46, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

• G001 — Use odd prime p=79, original a=46, 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 = [79]

derived = [[[2, 1], [3, 1], [4, 3], [5, 4], [6, 1], [7, 2], [8, 7]], Yes]

Euler ORIGINAL signed quotient times modulus := mul(0, 79)

Euler ORIGINAL signed quotient times modulus = 0

Euler ORIGINAL signed a minus modulus multiple := sub(46, 0)

Euler ORIGINAL signed a minus modulus multiple = 46

source roles = [46, 79, 0]

derived = [46, Yes]

source roles = [79, 46, 46]

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

Euler square root candidate61 := mul(61, 61)

Euler square root candidate61 = 3721

Euler square root candidate62 := mul(62, 62)

Euler square root candidate62 = 3844

Euler square root candidate63 := mul(63, 63)

Euler square root candidate63 = 3969

Euler square root candidate64 := mul(64, 64)

Euler square root candidate64 = 4096

Euler square root candidate65 := mul(65, 65)

Euler square root candidate65 = 4225

Euler square root candidate66 := mul(66, 66)

Euler square root candidate66 = 4356

Euler square root candidate67 := mul(67, 67)

Euler square root candidate67 = 4489

Euler square root candidate68 := mul(68, 68)

Euler square root candidate68 = 4624

Euler square root candidate69 := mul(69, 69)

Euler square root candidate69 = 4761

Euler square root candidate70 := mul(70, 70)

Euler square root candidate70 = 4900

Euler square root candidate71 := mul(71, 71)

Euler square root candidate71 = 5041

Euler square root candidate72 := mul(72, 72)

Euler square root candidate72 = 5184

Euler square root candidate73 := mul(73, 73)

Euler square root candidate73 = 5329

Euler square root candidate74 := mul(74, 74)

Euler square root candidate74 = 5476

Euler square root candidate75 := mul(75, 75)

Euler square root candidate75 = 5625

Euler square root candidate76 := mul(76, 76)

Euler square root candidate76 = 5776

Euler square root candidate77 := mul(77, 77)

Euler square root candidate77 = 5929

Euler square root candidate78 := mul(78, 78)

Euler square root candidate78 = 6084

source roles = [79, 46]

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

Euler binary power accumulate0 := mul(1, 46)

Euler binary power accumulate0 = 46

Euler binary power square0 := mul(46, 46)

Euler binary power square0 = 2116

source roles = [0, [1, 46, 39], 79, 46, 39]

derived = [46, 2116, 46, 62, 19]

Euler binary power accumulate1 := mul(46, 62)

Euler binary power accumulate1 = 2852

Euler binary power square1 := mul(62, 62)

Euler binary power square1 = 3844

source roles = [1, [46, 62, 19], 79, 46, 39]

derived = [2852, 3844, 8, 52, 9]

Euler binary power accumulate2 := mul(8, 52)

Euler binary power accumulate2 = 416

Euler binary power square2 := mul(52, 52)

Euler binary power square2 = 2704

source roles = [2, [8, 52, 9], 79, 46, 39]

derived = [416, 2704, 21, 18, 4]

Euler binary power square3 := mul(18, 18)

Euler binary power square3 = 324

source roles = [3, [21, 18, 4], 79, 46, 39]

derived = [null, 324, 21, 8, 2]

Euler binary power square4 := mul(8, 8)

Euler binary power square4 = 64

source roles = [4, [21, 8, 2], 79, 46, 39]

derived = [null, 64, 21, 64, 1]

Euler binary power accumulate5 := mul(21, 64)

Euler binary power accumulate5 = 1344

Euler binary power square5 := mul(64, 64)

Euler binary power square5 = 4096

source roles = [5, [21, 64, 1], 79, 46, 39]

derived = [1344, 4096, 1, 67, 0]

source roles = [79, [21, 58], 1, 1]

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

Reverse checks

• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Recompute Euler binary power square5 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Euler binary power accumulate5 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 binary power square4 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Recompute Euler binary power square3 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Recompute Euler binary power square2 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Euler binary power accumulate2 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Recompute Euler binary power square1 with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Euler binary power accumulate1 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 binary power square0 with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Recompute Euler binary power accumulate0 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S084 · premise_and_result_audit — Recompute Euler square root candidate78 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Euler square root candidate77 with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Euler square root candidate76 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Euler square root candidate75 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Euler square root candidate74 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Euler square root candidate73 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Euler square root candidate72 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Euler square root candidate71 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Euler square root candidate70 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Euler square root candidate69 with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Euler square root candidate68 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Euler square root candidate67 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Euler square root candidate66 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Euler square root candidate65 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Euler square root candidate64 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Euler square root candidate63 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Euler square root candidate62 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Euler square root candidate61 with exact rational arithmetic, not rounded decimals.
• 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

(79, 46, quadratic_residue, [21, 58], 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.