Step-by-step derivation
S001 · implies
Before
source operation=legendre-residue-nonresidue-witness;original snapshot and mathematics retained
After
explicit ordered source output fields=[prime, a, normalized_class, nonzero_roots, Legendre_symbol, 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 dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
Legendre ORIGINAL odd prime:ORIGINAL modulus=67;ALL possible trial divisors through square root=[2, 3, 4, 5, 6, 7, 8]
After
ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3], [5, 2], [6, 1], [7, 4], [8, 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 dependencies
Rule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S001
S003 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
ORIGINAL root x=1 in1..66;exact square=1
After
square=0*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S003
S005 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
ORIGINAL root x=2 in1..66;exact square=4
After
square=0*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S005
S007 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL root x=3 in1..66;exact square=9
After
square=0*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S007
S009 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
ORIGINAL root x=4 in1..66;exact square=16
After
square=0*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S009
S011 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
ORIGINAL root x=5 in1..66;exact square=25
After
square=0*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S011
S013 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
ORIGINAL root x=6 in1..66;exact square=36
After
square=0*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S013
S015 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · implies
Before
ORIGINAL root x=7 in1..66;exact square=49
After
square=0*67+49;canonical square class=49;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S015
S017 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · implies
Before
ORIGINAL root x=8 in1..66;exact square=64
After
square=0*67+64;canonical square class=64;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S017
S019 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
ORIGINAL root x=9 in1..66;exact square=81
After
square=1*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S019
S021 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · implies
Before
ORIGINAL root x=10 in1..66;exact square=100
After
square=1*67+33;canonical square class=33;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S021
S023 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · implies
Before
ORIGINAL root x=11 in1..66;exact square=121
After
square=1*67+54;canonical square class=54;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S023
S025 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · implies
Before
ORIGINAL root x=12 in1..66;exact square=144
After
square=2*67+10;canonical square class=10;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S025
S027 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · implies
Before
ORIGINAL root x=13 in1..66;exact square=169
After
square=2*67+35;canonical square class=35;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S027
S029 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · implies
Before
ORIGINAL root x=14 in1..66;exact square=196
After
square=2*67+62;canonical square class=62;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S029
S031 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · implies
Before
ORIGINAL root x=15 in1..66;exact square=225
After
square=3*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S031
S033 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · implies
Before
ORIGINAL root x=16 in1..66;exact square=256
After
square=3*67+55;canonical square class=55;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S033
S035 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · implies
Before
ORIGINAL root x=17 in1..66;exact square=289
After
square=4*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S035
S037 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · implies
Before
ORIGINAL root x=18 in1..66;exact square=324
After
square=4*67+56;canonical square class=56;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S037
S039 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · implies
Before
ORIGINAL root x=19 in1..66;exact square=361
After
square=5*67+26;canonical square class=26;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S039
S041 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S040
S042 · implies
Before
ORIGINAL root x=20 in1..66;exact square=400
After
square=5*67+65;canonical square class=65;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S041
S043 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S042
S044 · implies
Before
ORIGINAL root x=21 in1..66;exact square=441
After
square=6*67+39;canonical square class=39;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S043
S045 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S044
S046 · implies
Before
ORIGINAL root x=22 in1..66;exact square=484
After
square=7*67+15;canonical square class=15;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S045
S047 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S046
S048 · implies
Before
ORIGINAL root x=23 in1..66;exact square=529
After
square=7*67+60;canonical square class=60;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S047
S049 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · implies
Before
ORIGINAL root x=24 in1..66;exact square=576
After
square=8*67+40;canonical square class=40;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S049
S051 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S050
S052 · implies
Before
ORIGINAL root x=25 in1..66;exact square=625
After
square=9*67+22;canonical square class=22;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S051
S053 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S052
S054 · implies
Before
ORIGINAL root x=26 in1..66;exact square=676
After
square=10*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S053
S055 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S054
S056 · implies
Before
ORIGINAL root x=27 in1..66;exact square=729
After
square=10*67+59;canonical square class=59;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S055
S057 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S056
S058 · implies
Before
ORIGINAL root x=28 in1..66;exact square=784
After
square=11*67+47;canonical square class=47;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S057
S059 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S058
S060 · implies
Before
ORIGINAL root x=29 in1..66;exact square=841
After
square=12*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S059
S061 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S060
S062 · implies
Before
ORIGINAL root x=30 in1..66;exact square=900
After
square=13*67+29;canonical square class=29;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S061
S063 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S062
S064 · implies
Before
ORIGINAL root x=31 in1..66;exact square=961
After
square=14*67+23;canonical square class=23;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S063
S065 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S064
S066 · implies
Before
ORIGINAL root x=32 in1..66;exact square=1024
After
square=15*67+19;canonical square class=19;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S065
S067 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S066
S068 · implies
Before
ORIGINAL root x=33 in1..66;exact square=1089
After
square=16*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S067
S069 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S068
S070 · implies
Before
ORIGINAL root x=34 in1..66;exact square=1156
After
square=17*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S069
S071 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S070
S072 · implies
Before
ORIGINAL root x=35 in1..66;exact square=1225
After
square=18*67+19;canonical square class=19;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S071
S073 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S072
S074 · implies
Before
ORIGINAL root x=36 in1..66;exact square=1296
After
square=19*67+23;canonical square class=23;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S073
S075 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S074
S076 · implies
Before
ORIGINAL root x=37 in1..66;exact square=1369
After
square=20*67+29;canonical square class=29;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S075
S077 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S076
S078 · implies
Before
ORIGINAL root x=38 in1..66;exact square=1444
After
square=21*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S077
S079 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S078
S080 · implies
Before
ORIGINAL root x=39 in1..66;exact square=1521
After
square=22*67+47;canonical square class=47;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S079
S081 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S080
S082 · implies
Before
ORIGINAL root x=40 in1..66;exact square=1600
After
square=23*67+59;canonical square class=59;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S081
S083 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S082
S084 · implies
Before
ORIGINAL root x=41 in1..66;exact square=1681
After
square=25*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S083
S085 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S084
S086 · implies
Before
ORIGINAL root x=42 in1..66;exact square=1764
After
square=26*67+22;canonical square class=22;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S085
S087 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S086
S088 · implies
Before
ORIGINAL root x=43 in1..66;exact square=1849
After
square=27*67+40;canonical square class=40;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S087
S089 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S088
S090 · implies
Before
ORIGINAL root x=44 in1..66;exact square=1936
After
square=28*67+60;canonical square class=60;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S089
S091 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S090
S092 · implies
Before
ORIGINAL root x=45 in1..66;exact square=2025
After
square=30*67+15;canonical square class=15;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S091
S093 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S092
S094 · implies
Before
ORIGINAL root x=46 in1..66;exact square=2116
After
square=31*67+39;canonical square class=39;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S093
S095 · calculation
Before
Legendre ORIGINAL nonzero root47 exact square := mul(47, 47)
After
Legendre ORIGINAL nonzero root47 exact square = 2209
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S094
S096 · implies
Before
ORIGINAL root x=47 in1..66;exact square=2209
After
square=32*67+65;canonical square class=65;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S095
S097 · calculation
Before
Legendre ORIGINAL nonzero root48 exact square := mul(48, 48)
After
Legendre ORIGINAL nonzero root48 exact square = 2304
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S096
S098 · implies
Before
ORIGINAL root x=48 in1..66;exact square=2304
After
square=34*67+26;canonical square class=26;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S097
S099 · calculation
Before
Legendre ORIGINAL nonzero root49 exact square := mul(49, 49)
After
Legendre ORIGINAL nonzero root49 exact square = 2401
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S098
S100 · implies
Before
ORIGINAL root x=49 in1..66;exact square=2401
After
square=35*67+56;canonical square class=56;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S099
S101 · calculation
Before
Legendre ORIGINAL nonzero root50 exact square := mul(50, 50)
After
Legendre ORIGINAL nonzero root50 exact square = 2500
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S100
S102 · implies
Before
ORIGINAL root x=50 in1..66;exact square=2500
After
square=37*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S101
S103 · calculation
Before
Legendre ORIGINAL nonzero root51 exact square := mul(51, 51)
After
Legendre ORIGINAL nonzero root51 exact square = 2601
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S102
S104 · implies
Before
ORIGINAL root x=51 in1..66;exact square=2601
After
square=38*67+55;canonical square class=55;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S103
S105 · calculation
Before
Legendre ORIGINAL nonzero root52 exact square := mul(52, 52)
After
Legendre ORIGINAL nonzero root52 exact square = 2704
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S104
S106 · implies
Before
ORIGINAL root x=52 in1..66;exact square=2704
After
square=40*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S105
S107 · calculation
Before
Legendre ORIGINAL nonzero root53 exact square := mul(53, 53)
After
Legendre ORIGINAL nonzero root53 exact square = 2809
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S106
S108 · implies
Before
ORIGINAL root x=53 in1..66;exact square=2809
After
square=41*67+62;canonical square class=62;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S107
S109 · calculation
Before
Legendre ORIGINAL nonzero root54 exact square := mul(54, 54)
After
Legendre ORIGINAL nonzero root54 exact square = 2916
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S108
S110 · implies
Before
ORIGINAL root x=54 in1..66;exact square=2916
After
square=43*67+35;canonical square class=35;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S109
S111 · calculation
Before
Legendre ORIGINAL nonzero root55 exact square := mul(55, 55)
After
Legendre ORIGINAL nonzero root55 exact square = 3025
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S110
S112 · implies
Before
ORIGINAL root x=55 in1..66;exact square=3025
After
square=45*67+10;canonical square class=10;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S111
S113 · calculation
Before
Legendre ORIGINAL nonzero root56 exact square := mul(56, 56)
After
Legendre ORIGINAL nonzero root56 exact square = 3136
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S112
S114 · implies
Before
ORIGINAL root x=56 in1..66;exact square=3136
After
square=46*67+54;canonical square class=54;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S113
S115 · calculation
Before
Legendre ORIGINAL nonzero root57 exact square := mul(57, 57)
After
Legendre ORIGINAL nonzero root57 exact square = 3249
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S114
S116 · implies
Before
ORIGINAL root x=57 in1..66;exact square=3249
After
square=48*67+33;canonical square class=33;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S115
S117 · calculation
Before
Legendre ORIGINAL nonzero root58 exact square := mul(58, 58)
After
Legendre ORIGINAL nonzero root58 exact square = 3364
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S116
S118 · implies
Before
ORIGINAL root x=58 in1..66;exact square=3364
After
square=50*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S117
S119 · calculation
Before
Legendre ORIGINAL nonzero root59 exact square := mul(59, 59)
After
Legendre ORIGINAL nonzero root59 exact square = 3481
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S118
S120 · implies
Before
ORIGINAL root x=59 in1..66;exact square=3481
After
square=51*67+64;canonical square class=64;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S119
S121 · calculation
Before
Legendre ORIGINAL nonzero root60 exact square := mul(60, 60)
After
Legendre ORIGINAL nonzero root60 exact square = 3600
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S120
S122 · implies
Before
ORIGINAL root x=60 in1..66;exact square=3600
After
square=53*67+49;canonical square class=49;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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S121
S123 · calculation
Before
Legendre ORIGINAL nonzero root61 exact square := mul(61, 61)
After
Legendre ORIGINAL nonzero root61 exact square = 3721
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S122
S124 · implies
Before
ORIGINAL root x=61 in1..66;exact square=3721
After
square=55*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S123
S125 · calculation
Before
Legendre ORIGINAL nonzero root62 exact square := mul(62, 62)
After
Legendre ORIGINAL nonzero root62 exact square = 3844
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S124
S126 · implies
Before
ORIGINAL root x=62 in1..66;exact square=3844
After
square=57*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S125
S127 · calculation
Before
Legendre ORIGINAL nonzero root63 exact square := mul(63, 63)
After
Legendre ORIGINAL nonzero root63 exact square = 3969
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S126
S128 · implies
Before
ORIGINAL root x=63 in1..66;exact square=3969
After
square=59*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S127
S129 · calculation
Before
Legendre ORIGINAL nonzero root64 exact square := mul(64, 64)
After
Legendre ORIGINAL nonzero root64 exact square = 4096
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S128
S130 · implies
Before
ORIGINAL root x=64 in1..66;exact square=4096
After
square=61*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S129
S131 · calculation
Before
Legendre ORIGINAL nonzero root65 exact square := mul(65, 65)
After
Legendre ORIGINAL nonzero root65 exact square = 4225
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S130
S132 · implies
Before
ORIGINAL root x=65 in1..66;exact square=4225
After
square=63*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S131
S133 · calculation
Before
Legendre ORIGINAL nonzero root66 exact square := mul(66, 66)
After
Legendre ORIGINAL nonzero root66 exact square = 4356
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S132
S134 · implies
Before
ORIGINAL root x=66 in1..66;exact square=4356
After
square=65*67+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 dependencies
Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S133
S135 · implies
Before
ORIGINAL prime=67;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 14], [10, 33], [11, 54], [12, 10], [13, 35], [14, 62], [15, 24], [16, 55], [17, 21], [18, 56], [19, 26], [20, 65], [21, 39], [22, 15], [23, 60], [24, 40], [25, 22], [26, 6], [27, 59], [28, 47], [29, 37], [30, 29], [31, 23], [32, 19], [33, 17], [34, 17], [35, 19], [36, 23], [37, 29], [38, 37], [39, 47], [40, 59], [41, 6], [42, 22], [43, 40], [44, 60], [45, 15], [46, 39], [47, 65], [48, 26], [49, 56], [50, 21], [51, 55], [52, 24], [53, 62], [54, 35], [55, 10], [56, 54], [57, 33], [58, 14], [59, 64], [60, 49], [61, 36], [62, 25], [63, 16], [64, 9], [65, 4], [66, 1]];nonzero square classes=[1, 4, 6, 9, 10, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 29, 33, 35, 36, 37, 39, 40, 47, 49, 54, 55, 56, 59, 60, 62, 64, 65]
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 dependencies
Rule: legendre_ZERO_then_residue_nonresidue_three_way_definition · Uses: G001, S134
S136 · calculation
Before
Legendre ORIGINAL candidate class quotient times modulus := mul(0, 67)
After
Legendre ORIGINAL candidate class quotient times modulus = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S135
S137 · calculation
Before
Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(18, 0)
After
Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple = 18
Conditions: none
Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S136
S138 · implies
Before
Legendre ORIGINAL candidate class:ORIGINAL a=18,positive modulus=67;actual quotient=0
After
normalized class=18;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 dependencies
Rule: legendre_ORIGINAL_signed_class_Euclidean_normalization · Uses: G001, S137