# Question 23789: Divisibility by p Gives Symbol Zero, Not Plus One [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-0064) 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. R0046-23789 · q-20261014-advance-20261112-r06-legendre-symbol-0064 ## Original question and givens Use odd prime p=47, original a=658. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate). - G001 — Use odd prime p=47, original a=658. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, 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-zero-class;original snapshot and mathematics retained After explicit ordered source output fields=[prime, a, normalized_class, Legendre_symbol, nonzero_roots, 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=47;ALL possible trial divisors through square root=[2, 3, 4, 5, 6] After ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 3], [5, 2], [6, 5]];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..46;exact square=1 After square=0*47+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..46;exact square=4 After square=0*47+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..46;exact square=9 After square=0*47+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..46;exact square=16 After square=0*47+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..46;exact square=25 After square=0*47+25;canonical square class=25;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..46;exact square=36 After square=0*47+36;canonical square class=36;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..46;exact square=49 After square=1*47+2;canonical square class=2;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..46;exact square=64 After square=1*47+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, 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..46;exact square=81 After square=1*47+34;canonical square class=34;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..46;exact square=100 After square=2*47+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, 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..46;exact square=121 After square=2*47+27;canonical square class=27;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..46;exact square=144 After square=3*47+3;canonical square class=3;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..46;exact square=169 After square=3*47+28;canonical square class=28;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..46;exact square=196 After square=4*47+8;canonical square class=8;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..46;exact square=225 After square=4*47+37;canonical square class=37;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..46;exact square=256 After square=5*47+21;canonical square class=21;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..46;exact square=289 After square=6*47+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, 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..46;exact square=324 After square=6*47+42;canonical square class=42;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 · calculationBefore Legendre ORIGINAL nonzero root19 exact square := mul(19, 19) After Legendre ORIGINAL nonzero root19 exact square = 361 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, S038 S040 · impliesBefore ORIGINAL root x=19 in1..46;exact square=361 After square=7*47+32;canonical square class=32;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, S039 S041 · calculationBefore Legendre ORIGINAL nonzero root20 exact square := mul(20, 20) After Legendre ORIGINAL nonzero root20 exact square = 400 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, S040 S042 · impliesBefore ORIGINAL root x=20 in1..46;exact square=400 After square=8*47+24;canonical square class=24;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, S041 S043 · calculationBefore Legendre ORIGINAL nonzero root21 exact square := mul(21, 21) After Legendre ORIGINAL nonzero root21 exact square = 441 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, S042 S044 · impliesBefore ORIGINAL root x=21 in1..46;exact square=441 After square=9*47+18;canonical square class=18;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, S043 S045 · calculationBefore Legendre ORIGINAL nonzero root22 exact square := mul(22, 22) After Legendre ORIGINAL nonzero root22 exact square = 484 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, S044 S046 · impliesBefore ORIGINAL root x=22 in1..46;exact square=484 After square=10*47+14;canonical square class=14;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, S045 S047 · calculationBefore Legendre ORIGINAL nonzero root23 exact square := mul(23, 23) After Legendre ORIGINAL nonzero root23 exact square = 529 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, S046 S048 · impliesBefore ORIGINAL root x=23 in1..46;exact square=529 After square=11*47+12;canonical square class=12;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, S047 S049 · calculationBefore Legendre ORIGINAL nonzero root24 exact square := mul(24, 24) After Legendre ORIGINAL nonzero root24 exact square = 576 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, S048 S050 · impliesBefore ORIGINAL root x=24 in1..46;exact square=576 After square=12*47+12;canonical square class=12;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, S049 S051 · calculationBefore Legendre ORIGINAL nonzero root25 exact square := mul(25, 25) After Legendre ORIGINAL nonzero root25 exact square = 625 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, S050 S052 · impliesBefore ORIGINAL root x=25 in1..46;exact square=625 After square=13*47+14;canonical square class=14;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, S051 S053 · calculationBefore Legendre ORIGINAL nonzero root26 exact square := mul(26, 26) After Legendre ORIGINAL nonzero root26 exact square = 676 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, S052 S054 · impliesBefore ORIGINAL root x=26 in1..46;exact square=676 After square=14*47+18;canonical square class=18;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, S053 S055 · calculationBefore Legendre ORIGINAL nonzero root27 exact square := mul(27, 27) After Legendre ORIGINAL nonzero root27 exact square = 729 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, S054 S056 · impliesBefore ORIGINAL root x=27 in1..46;exact square=729 After square=15*47+24;canonical square class=24;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, S055 S057 · calculationBefore Legendre ORIGINAL nonzero root28 exact square := mul(28, 28) After Legendre ORIGINAL nonzero root28 exact square = 784 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, S056 S058 · impliesBefore ORIGINAL root x=28 in1..46;exact square=784 After square=16*47+32;canonical square class=32;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, S057 S059 · calculationBefore Legendre ORIGINAL nonzero root29 exact square := mul(29, 29) After Legendre ORIGINAL nonzero root29 exact square = 841 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, S058 S060 · impliesBefore ORIGINAL root x=29 in1..46;exact square=841 After square=17*47+42;canonical square class=42;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, S059 S061 · calculationBefore Legendre ORIGINAL nonzero root30 exact square := mul(30, 30) After Legendre ORIGINAL nonzero root30 exact square = 900 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, S060 S062 · impliesBefore ORIGINAL root x=30 in1..46;exact square=900 After square=19*47+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, S061 S063 · calculationBefore Legendre ORIGINAL nonzero root31 exact square := mul(31, 31) After Legendre ORIGINAL nonzero root31 exact square = 961 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, S062 S064 · impliesBefore ORIGINAL root x=31 in1..46;exact square=961 After square=20*47+21;canonical square class=21;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, S063 S065 · calculationBefore Legendre ORIGINAL nonzero root32 exact square := mul(32, 32) After Legendre ORIGINAL nonzero root32 exact square = 1024 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, S064 S066 · impliesBefore ORIGINAL root x=32 in1..46;exact square=1024 After square=21*47+37;canonical square class=37;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, S065 S067 · calculationBefore Legendre ORIGINAL nonzero root33 exact square := mul(33, 33) After Legendre ORIGINAL nonzero root33 exact square = 1089 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, S066 S068 · impliesBefore ORIGINAL root x=33 in1..46;exact square=1089 After square=23*47+8;canonical square class=8;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, S067 S069 · calculationBefore Legendre ORIGINAL nonzero root34 exact square := mul(34, 34) After Legendre ORIGINAL nonzero root34 exact square = 1156 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, S068 S070 · impliesBefore ORIGINAL root x=34 in1..46;exact square=1156 After square=24*47+28;canonical square class=28;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, S069 S071 · calculationBefore Legendre ORIGINAL nonzero root35 exact square := mul(35, 35) After Legendre ORIGINAL nonzero root35 exact square = 1225 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, S070 S072 · impliesBefore ORIGINAL root x=35 in1..46;exact square=1225 After square=26*47+3;canonical square class=3;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, S071 S073 · calculationBefore Legendre ORIGINAL nonzero root36 exact square := mul(36, 36) After Legendre ORIGINAL nonzero root36 exact square = 1296 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, S072 S074 · impliesBefore ORIGINAL root x=36 in1..46;exact square=1296 After square=27*47+27;canonical square class=27;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, S073 S075 · calculationBefore Legendre ORIGINAL nonzero root37 exact square := mul(37, 37) After Legendre ORIGINAL nonzero root37 exact square = 1369 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, S074 S076 · impliesBefore ORIGINAL root x=37 in1..46;exact square=1369 After square=29*47+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, S075 S077 · calculationBefore Legendre ORIGINAL nonzero root38 exact square := mul(38, 38) After Legendre ORIGINAL nonzero root38 exact square = 1444 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, S076 S078 · impliesBefore ORIGINAL root x=38 in1..46;exact square=1444 After square=30*47+34;canonical square class=34;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, S077 S079 · calculationBefore Legendre ORIGINAL nonzero root39 exact square := mul(39, 39) After Legendre ORIGINAL nonzero root39 exact square = 1521 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, S078 S080 · impliesBefore ORIGINAL root x=39 in1..46;exact square=1521 After square=32*47+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, S079 S081 · calculationBefore Legendre ORIGINAL nonzero root40 exact square := mul(40, 40) After Legendre ORIGINAL nonzero root40 exact square = 1600 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, S080 S082 · impliesBefore ORIGINAL root x=40 in1..46;exact square=1600 After square=34*47+2;canonical square class=2;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, S081 S083 · calculationBefore Legendre ORIGINAL nonzero root41 exact square := mul(41, 41) After Legendre ORIGINAL nonzero root41 exact square = 1681 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, S082 S084 · impliesBefore ORIGINAL root x=41 in1..46;exact square=1681 After square=35*47+36;canonical square class=36;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, S083 S085 · calculationBefore Legendre ORIGINAL nonzero root42 exact square := mul(42, 42) After Legendre ORIGINAL nonzero root42 exact square = 1764 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, S084 S086 · impliesBefore ORIGINAL root x=42 in1..46;exact square=1764 After square=37*47+25;canonical square class=25;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, S085 S087 · calculationBefore Legendre ORIGINAL nonzero root43 exact square := mul(43, 43) After Legendre ORIGINAL nonzero root43 exact square = 1849 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, S086 S088 · impliesBefore ORIGINAL root x=43 in1..46;exact square=1849 After square=39*47+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, S087 S089 · calculationBefore Legendre ORIGINAL nonzero root44 exact square := mul(44, 44) After Legendre ORIGINAL nonzero root44 exact square = 1936 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, S088 S090 · impliesBefore ORIGINAL root x=44 in1..46;exact square=1936 After square=41*47+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, S089 S091 · calculationBefore Legendre ORIGINAL nonzero root45 exact square := mul(45, 45) After Legendre ORIGINAL nonzero root45 exact square = 2025 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, S090 S092 · impliesBefore ORIGINAL root x=45 in1..46;exact square=2025 After square=43*47+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, S091 S093 · calculationBefore Legendre ORIGINAL nonzero root46 exact square := mul(46, 46) After Legendre ORIGINAL nonzero root46 exact square = 2116 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, S092 S094 · impliesBefore ORIGINAL root x=46 in1..46;exact square=2116 After square=45*47+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, S093 S095 · impliesBefore ORIGINAL prime=47;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 2], [8, 17], [9, 34], [10, 6], [11, 27], [12, 3], [13, 28], [14, 8], [15, 37], [16, 21], [17, 7], [18, 42], [19, 32], [20, 24], [21, 18], [22, 14], [23, 12], [24, 12], [25, 14], [26, 18], [27, 24], [28, 32], [29, 42], [30, 7], [31, 21], [32, 37], [33, 8], [34, 28], [35, 3], [36, 27], [37, 6], [38, 34], [39, 17], [40, 2], [41, 36], [42, 25], [43, 16], [44, 9], [45, 4], [46, 1]];nonzero square classes=[1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 17, 18, 21, 24, 25, 27, 28, 32, 34, 36, 37, 42] 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, S094 S096 · calculationBefore Legendre ORIGINAL candidate class quotient times modulus := mul(14, 47) After Legendre ORIGINAL candidate class quotient times modulus = 658 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, S095 S097 · calculationBefore Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(658, 658) After Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple = 0 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, S096 S098 · impliesBefore Legendre ORIGINAL candidate class:ORIGINAL a=658,positive modulus=47;actual quotient=14 After normalized class=0;a=quotient*modulus+class;ORIGINAL a retained in output where declared Conditions: none Retain the same class for negative a and normalize into0..p−1,rather than using a negative language-specific remainder. Step identifiers and dependenciesRule: legendre_ORIGINAL_signed_class_Euclidean_normalization · Uses: G001, S097 ## Reverse checks - 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 Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute Legendre ORIGINAL candidate class quotient times modulus with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 Legendre ORIGINAL nonzero root46 exact square with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S091 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root45 exact square with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S089 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root44 exact square 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 Legendre ORIGINAL nonzero root43 exact square with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S085 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root42 exact square with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S083 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root41 exact square with exact rational arithmetic, not rounded decimals. - 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 — Recompute Legendre ORIGINAL nonzero root40 exact square with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S079 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root39 exact square 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 Legendre ORIGINAL nonzero root38 exact square with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root37 exact square with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root36 exact square 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 Legendre ORIGINAL nonzero root35 exact square with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S069 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root34 exact square with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S067 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root33 exact square with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root32 exact square with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root31 exact square with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root30 exact square with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root29 exact square with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root28 exact square with exact rational arithmetic, not rounded decimals. - 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 — Recompute Legendre ORIGINAL nonzero root27 exact square with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root26 exact square 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 Legendre ORIGINAL nonzero root25 exact square 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 Legendre ORIGINAL nonzero root24 exact square 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 Legendre ORIGINAL nonzero root23 exact square with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root22 exact square with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root21 exact square with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root20 exact square with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root19 exact square with exact rational arithmetic, not rounded decimals. - 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 (47, 658, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification) 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: 8099cf413de353bd2267ce872539ff406ba37d1cdadcf63aea6f2c751fba838e - candidate record SHA-256: a16ea14c7427045fffcf800c5b1aaf98ecb83f53be1505bcade15123f4055b75 - 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-0064.json)