Question 24186: Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion

Original question and givens

Use odd prime p=37, original a=10, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).

• G001 — Use odd prime p=37, original a=10, 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 = [37]

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

Euler ORIGINAL signed quotient times modulus := mul(0, 37)

Euler ORIGINAL signed quotient times modulus = 0

Euler ORIGINAL signed a minus modulus multiple := sub(10, 0)

Euler ORIGINAL signed a minus modulus multiple = 10

source roles = [10, 37, 0]

derived = [10, Yes]

source roles = [37, 10, 10]

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

source roles = [37, 10]

derived = [[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 12], [8, 27], [9, 7], [10, 26], [11, 10], [12, 33], [13, 21], [14, 11], [15, 3], [16, 34], [17, 30], [18, 28], [19, 28], [20, 30], [21, 34], [22, 3], [23, 11], [24, 21], [25, 33], [26, 10], [27, 26], [28, 7], [29, 27], [30, 12], [31, 36], [32, 25], [33, 16], [34, 9], [35, 4], [36, 1]], [11, 26], 1]

Euler binary power square0 := mul(10, 10)

Euler binary power square0 = 100

source roles = [0, [1, 10, 18], 37, 10, 18]

derived = [null, 100, 1, 26, 9]

Euler binary power accumulate1 := mul(1, 26)

Euler binary power accumulate1 = 26

Euler binary power square1 := mul(26, 26)

Euler binary power square1 = 676

source roles = [1, [1, 26, 9], 37, 10, 18]

derived = [26, 676, 26, 10, 4]

Euler binary power square2 := mul(10, 10)

Euler binary power square2 = 100

source roles = [2, [26, 10, 4], 37, 10, 18]

derived = [null, 100, 26, 26, 2]

Euler binary power square3 := mul(26, 26)

Euler binary power square3 = 676

source roles = [3, [26, 26, 2], 37, 10, 18]

derived = [null, 676, 26, 10, 1]

Euler binary power accumulate4 := mul(26, 10)

Euler binary power accumulate4 = 260

Euler binary power square4 := mul(10, 10)

Euler binary power square4 = 100

source roles = [4, [26, 10, 1], 37, 10, 18]

derived = [260, 100, 1, 26, 0]

source roles = [37, [11, 26], 1, 1]

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

Reverse checks

• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Recompute Euler binary power square4 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Euler binary power accumulate4 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Recompute Euler binary power square3 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Recompute Euler binary power square2 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 binary power square1 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Euler binary power accumulate1 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Recompute Euler binary power square0 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 — 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

(37, 10, quadratic_residue, [11, 26], 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.