# Question 23707: Legendre Classes Include Zero before Nonzero Residue Classification [Return to catalogue page 119](https://ocme.evemisslab.com/en/questions/page/119/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261112-r06-legendre-symbol-foundations/#q-20261014-advance-20261112-r06-legendre-symbol-0007) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0046-23707 · q-20261014-advance-20261112-r06-legendre-symbol-0007 ## Original question and givens Use bounded odd prime p=19 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate). - G001 — Use bounded odd prime p=19 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, 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-legendre-symbol](https://ocme.evemisslab.com/en/mko/mko-legendre-symbol/) ## Step-by-step derivation S001 · impliesBefore source operation=legendre-classification-table;original snapshot and mathematics retained After explicit ordered source output fields=[prime, class_symbol_rows, complete_partition, certificate] Conditions: none Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore Legendre ORIGINAL odd prime:ORIGINAL modulus=19;ALL possible trial divisors through square root=[2, 3, 4] After ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3]];prime=Yes Conditions: none Qualify primality over the full trial range,not by guessing from Legendre/Euler classification.This is a modulus premise,not the square status of a. Step identifiers and dependenciesRule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S001 S003 · calculationBefore Legendre ORIGINAL nonzero root1 exact square := mul(1, 1) After Legendre ORIGINAL nonzero root1 exact square = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore ORIGINAL root x=1 in1..18;exact square=1 After square=0*19+1;canonical square class=1;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S003 S005 · calculationBefore Legendre ORIGINAL nonzero root2 exact square := mul(2, 2) After Legendre ORIGINAL nonzero root2 exact square = 4 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore ORIGINAL root x=2 in1..18;exact square=4 After square=0*19+4;canonical square class=4;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S005 S007 · calculationBefore Legendre ORIGINAL nonzero root3 exact square := mul(3, 3) After Legendre ORIGINAL nonzero root3 exact square = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL root x=3 in1..18;exact square=9 After square=0*19+9;canonical square class=9;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S007 S009 · calculationBefore Legendre ORIGINAL nonzero root4 exact square := mul(4, 4) After Legendre ORIGINAL nonzero root4 exact square = 16 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore ORIGINAL root x=4 in1..18;exact square=16 After square=0*19+16;canonical square class=16;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S009 S011 · calculationBefore Legendre ORIGINAL nonzero root5 exact square := mul(5, 5) After Legendre ORIGINAL nonzero root5 exact square = 25 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore ORIGINAL root x=5 in1..18;exact square=25 After square=1*19+6;canonical square class=6;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S011 S013 · calculationBefore Legendre ORIGINAL nonzero root6 exact square := mul(6, 6) After Legendre ORIGINAL nonzero root6 exact square = 36 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore ORIGINAL root x=6 in1..18;exact square=36 After square=1*19+17;canonical square class=17;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S013 S015 · calculationBefore Legendre ORIGINAL nonzero root7 exact square := mul(7, 7) After Legendre ORIGINAL nonzero root7 exact square = 49 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore ORIGINAL root x=7 in1..18;exact square=49 After square=2*19+11;canonical square class=11;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S015 S017 · calculationBefore Legendre ORIGINAL nonzero root8 exact square := mul(8, 8) After Legendre ORIGINAL nonzero root8 exact square = 64 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore ORIGINAL root x=8 in1..18;exact square=64 After square=3*19+7;canonical square class=7;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S017 S019 · calculationBefore Legendre ORIGINAL nonzero root9 exact square := mul(9, 9) After Legendre ORIGINAL nonzero root9 exact square = 81 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore ORIGINAL root x=9 in1..18;exact square=81 After square=4*19+5;canonical square class=5;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S019 S021 · calculationBefore Legendre ORIGINAL nonzero root10 exact square := mul(10, 10) After Legendre ORIGINAL nonzero root10 exact square = 100 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore ORIGINAL root x=10 in1..18;exact square=100 After square=5*19+5;canonical square class=5;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S021 S023 · calculationBefore Legendre ORIGINAL nonzero root11 exact square := mul(11, 11) After Legendre ORIGINAL nonzero root11 exact square = 121 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore ORIGINAL root x=11 in1..18;exact square=121 After square=6*19+7;canonical square class=7;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S023 S025 · calculationBefore Legendre ORIGINAL nonzero root12 exact square := mul(12, 12) After Legendre ORIGINAL nonzero root12 exact square = 144 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore ORIGINAL root x=12 in1..18;exact square=144 After square=7*19+11;canonical square class=11;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S025 S027 · calculationBefore Legendre ORIGINAL nonzero root13 exact square := mul(13, 13) After Legendre ORIGINAL nonzero root13 exact square = 169 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore ORIGINAL root x=13 in1..18;exact square=169 After square=8*19+17;canonical square class=17;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S027 S029 · calculationBefore Legendre ORIGINAL nonzero root14 exact square := mul(14, 14) After Legendre ORIGINAL nonzero root14 exact square = 196 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore ORIGINAL root x=14 in1..18;exact square=196 After square=10*19+6;canonical square class=6;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S029 S031 · calculationBefore Legendre ORIGINAL nonzero root15 exact square := mul(15, 15) After Legendre ORIGINAL nonzero root15 exact square = 225 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore ORIGINAL root x=15 in1..18;exact square=225 After square=11*19+16;canonical square class=16;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S031 S033 · calculationBefore Legendre ORIGINAL nonzero root16 exact square := mul(16, 16) After Legendre ORIGINAL nonzero root16 exact square = 256 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore ORIGINAL root x=16 in1..18;exact square=256 After square=13*19+9;canonical square class=9;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S033 S035 · calculationBefore Legendre ORIGINAL nonzero root17 exact square := mul(17, 17) After Legendre ORIGINAL nonzero root17 exact square = 289 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · impliesBefore ORIGINAL root x=17 in1..18;exact square=289 After square=15*19+4;canonical square class=4;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S035 S037 · calculationBefore Legendre ORIGINAL nonzero root18 exact square := mul(18, 18) After Legendre ORIGINAL nonzero root18 exact square = 324 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore ORIGINAL root x=18 in1..18;exact square=324 After square=17*19+1;canonical square class=1;ALL root indices retained Conditions: none Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used. Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S037 S039 · impliesBefore ORIGINAL prime=19;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 6], [6, 17], [7, 11], [8, 7], [9, 5], [10, 5], [11, 7], [12, 11], [13, 17], [14, 6], [15, 16], [16, 9], [17, 4], [18, 1]];nonzero square classes=[1, 4, 5, 6, 7, 9, 11, 16, 17] After for EACH normalized class r,first r0 ->symbol0;else square-table member ->symbol1;otherwise ->symbol-1;these cases partition ALL0..p-1 Conditions: none Separate zero first;the fact that0 is a square does not give symbol+1.Nonzero classes are classified by the complete square table.This is definition plus exhaustive evidence,not a later multiplicativity or Euler formula. Step identifiers and dependenciesRule: legendre_ZERO_then_residue_nonresidue_three_way_definition · Uses: G001, S038 ## Reverse checks - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root18 exact square with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root17 exact square with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root16 exact square with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root15 exact square with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root14 exact square with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root13 exact square with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root12 exact square with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root11 exact square with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root10 exact square with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root9 exact square with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root8 exact square with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root7 exact square with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root6 exact square with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root5 exact square with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root4 exact square with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root3 exact square 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 — Recompute Legendre ORIGINAL nonzero root2 exact square with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root1 exact square 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 (19, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1]], Yes, zero_residue_nonresidue_partition_is_complete) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: b8442f7b4cb82a93c40811d3ea0f00e8caa198e1ba43dc445b775d31ae477098 - candidate record SHA-256: 601731a74281f819e32fe6db965a3132af2210b741e6450ebc22184aa8087290 - semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1 - question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r06-legendre-symbol-0007.json)