Step-by-step derivation
S001 · implies
Before
source operation=quadratic-paired-roots;original snapshot and mathematics retained
After
explicit ordered source output fields=[p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, certificate]
Conditions: none
Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
Quadratic ORIGINAL modulus:ORIGINAL integer=67;complete possible trial divisors=[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;a composite>1 has a factor at most its square root
Conditions: none
Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification.
Step identifiers and dependencies
Rule: finite_prime_ALL_trial_divisors_through_square_root · Uses: G001, S001
S003 · calculation
Before
Quadratic ORIGINAL nonzero root1 square := mul(1, 1)
After
Quadratic ORIGINAL nonzero root1 square = 1
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
ORIGINAL nonzero root x=1 in1..66;actual square=1
After
Euclidean square=0*67+1;nonzero square class=1;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S003
S005 · calculation
Before
Quadratic ORIGINAL nonzero root2 square := mul(2, 2)
After
Quadratic ORIGINAL nonzero root2 square = 4
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
ORIGINAL nonzero root x=2 in1..66;actual square=4
After
Euclidean square=0*67+4;nonzero square class=4;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S005
S007 · calculation
Before
Quadratic ORIGINAL nonzero root3 square := mul(3, 3)
After
Quadratic ORIGINAL nonzero root3 square = 9
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL nonzero root x=3 in1..66;actual square=9
After
Euclidean square=0*67+9;nonzero square class=9;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S007
S009 · calculation
Before
Quadratic ORIGINAL nonzero root4 square := mul(4, 4)
After
Quadratic ORIGINAL nonzero root4 square = 16
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
ORIGINAL nonzero root x=4 in1..66;actual square=16
After
Euclidean square=0*67+16;nonzero square class=16;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S009
S011 · calculation
Before
Quadratic ORIGINAL nonzero root5 square := mul(5, 5)
After
Quadratic ORIGINAL nonzero root5 square = 25
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
ORIGINAL nonzero root x=5 in1..66;actual square=25
After
Euclidean square=0*67+25;nonzero square class=25;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S011
S013 · calculation
Before
Quadratic ORIGINAL nonzero root6 square := mul(6, 6)
After
Quadratic ORIGINAL nonzero root6 square = 36
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
ORIGINAL nonzero root x=6 in1..66;actual square=36
After
Euclidean square=0*67+36;nonzero square class=36;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S013
S015 · calculation
Before
Quadratic ORIGINAL nonzero root7 square := mul(7, 7)
After
Quadratic ORIGINAL nonzero root7 square = 49
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · implies
Before
ORIGINAL nonzero root x=7 in1..66;actual square=49
After
Euclidean square=0*67+49;nonzero square class=49;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S015
S017 · calculation
Before
Quadratic ORIGINAL nonzero root8 square := mul(8, 8)
After
Quadratic ORIGINAL nonzero root8 square = 64
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · implies
Before
ORIGINAL nonzero root x=8 in1..66;actual square=64
After
Euclidean square=0*67+64;nonzero square class=64;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S017
S019 · calculation
Before
Quadratic ORIGINAL nonzero root9 square := mul(9, 9)
After
Quadratic ORIGINAL nonzero root9 square = 81
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
ORIGINAL nonzero root x=9 in1..66;actual square=81
After
Euclidean square=1*67+14;nonzero square class=14;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S019
S021 · calculation
Before
Quadratic ORIGINAL nonzero root10 square := mul(10, 10)
After
Quadratic ORIGINAL nonzero root10 square = 100
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · implies
Before
ORIGINAL nonzero root x=10 in1..66;actual square=100
After
Euclidean square=1*67+33;nonzero square class=33;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S021
S023 · calculation
Before
Quadratic ORIGINAL nonzero root11 square := mul(11, 11)
After
Quadratic ORIGINAL nonzero root11 square = 121
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · implies
Before
ORIGINAL nonzero root x=11 in1..66;actual square=121
After
Euclidean square=1*67+54;nonzero square class=54;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S023
S025 · calculation
Before
Quadratic ORIGINAL nonzero root12 square := mul(12, 12)
After
Quadratic ORIGINAL nonzero root12 square = 144
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · implies
Before
ORIGINAL nonzero root x=12 in1..66;actual square=144
After
Euclidean square=2*67+10;nonzero square class=10;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S025
S027 · calculation
Before
Quadratic ORIGINAL nonzero root13 square := mul(13, 13)
After
Quadratic ORIGINAL nonzero root13 square = 169
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · implies
Before
ORIGINAL nonzero root x=13 in1..66;actual square=169
After
Euclidean square=2*67+35;nonzero square class=35;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S027
S029 · calculation
Before
Quadratic ORIGINAL nonzero root14 square := mul(14, 14)
After
Quadratic ORIGINAL nonzero root14 square = 196
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · implies
Before
ORIGINAL nonzero root x=14 in1..66;actual square=196
After
Euclidean square=2*67+62;nonzero square class=62;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S029
S031 · calculation
Before
Quadratic ORIGINAL nonzero root15 square := mul(15, 15)
After
Quadratic ORIGINAL nonzero root15 square = 225
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · implies
Before
ORIGINAL nonzero root x=15 in1..66;actual square=225
After
Euclidean square=3*67+24;nonzero square class=24;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S031
S033 · calculation
Before
Quadratic ORIGINAL nonzero root16 square := mul(16, 16)
After
Quadratic ORIGINAL nonzero root16 square = 256
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · implies
Before
ORIGINAL nonzero root x=16 in1..66;actual square=256
After
Euclidean square=3*67+55;nonzero square class=55;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S033
S035 · calculation
Before
Quadratic ORIGINAL nonzero root17 square := mul(17, 17)
After
Quadratic ORIGINAL nonzero root17 square = 289
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · implies
Before
ORIGINAL nonzero root x=17 in1..66;actual square=289
After
Euclidean square=4*67+21;nonzero square class=21;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S035
S037 · calculation
Before
Quadratic ORIGINAL nonzero root18 square := mul(18, 18)
After
Quadratic ORIGINAL nonzero root18 square = 324
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · implies
Before
ORIGINAL nonzero root x=18 in1..66;actual square=324
After
Euclidean square=4*67+56;nonzero square class=56;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S037
S039 · calculation
Before
Quadratic ORIGINAL nonzero root19 square := mul(19, 19)
After
Quadratic ORIGINAL nonzero root19 square = 361
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · implies
Before
ORIGINAL nonzero root x=19 in1..66;actual square=361
After
Euclidean square=5*67+26;nonzero square class=26;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S039
S041 · calculation
Before
Quadratic ORIGINAL nonzero root20 square := mul(20, 20)
After
Quadratic ORIGINAL nonzero root20 square = 400
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S040
S042 · implies
Before
ORIGINAL nonzero root x=20 in1..66;actual square=400
After
Euclidean square=5*67+65;nonzero square class=65;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S041
S043 · calculation
Before
Quadratic ORIGINAL nonzero root21 square := mul(21, 21)
After
Quadratic ORIGINAL nonzero root21 square = 441
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S042
S044 · implies
Before
ORIGINAL nonzero root x=21 in1..66;actual square=441
After
Euclidean square=6*67+39;nonzero square class=39;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S043
S045 · calculation
Before
Quadratic ORIGINAL nonzero root22 square := mul(22, 22)
After
Quadratic ORIGINAL nonzero root22 square = 484
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S044
S046 · implies
Before
ORIGINAL nonzero root x=22 in1..66;actual square=484
After
Euclidean square=7*67+15;nonzero square class=15;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S045
S047 · calculation
Before
Quadratic ORIGINAL nonzero root23 square := mul(23, 23)
After
Quadratic ORIGINAL nonzero root23 square = 529
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S046
S048 · implies
Before
ORIGINAL nonzero root x=23 in1..66;actual square=529
After
Euclidean square=7*67+60;nonzero square class=60;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S047
S049 · calculation
Before
Quadratic ORIGINAL nonzero root24 square := mul(24, 24)
After
Quadratic ORIGINAL nonzero root24 square = 576
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · implies
Before
ORIGINAL nonzero root x=24 in1..66;actual square=576
After
Euclidean square=8*67+40;nonzero square class=40;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S049
S051 · calculation
Before
Quadratic ORIGINAL nonzero root25 square := mul(25, 25)
After
Quadratic ORIGINAL nonzero root25 square = 625
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S050
S052 · implies
Before
ORIGINAL nonzero root x=25 in1..66;actual square=625
After
Euclidean square=9*67+22;nonzero square class=22;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S051
S053 · calculation
Before
Quadratic ORIGINAL nonzero root26 square := mul(26, 26)
After
Quadratic ORIGINAL nonzero root26 square = 676
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S052
S054 · implies
Before
ORIGINAL nonzero root x=26 in1..66;actual square=676
After
Euclidean square=10*67+6;nonzero square class=6;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S053
S055 · calculation
Before
Quadratic ORIGINAL nonzero root27 square := mul(27, 27)
After
Quadratic ORIGINAL nonzero root27 square = 729
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S054
S056 · implies
Before
ORIGINAL nonzero root x=27 in1..66;actual square=729
After
Euclidean square=10*67+59;nonzero square class=59;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S055
S057 · calculation
Before
Quadratic ORIGINAL nonzero root28 square := mul(28, 28)
After
Quadratic ORIGINAL nonzero root28 square = 784
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S056
S058 · implies
Before
ORIGINAL nonzero root x=28 in1..66;actual square=784
After
Euclidean square=11*67+47;nonzero square class=47;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S057
S059 · calculation
Before
Quadratic ORIGINAL nonzero root29 square := mul(29, 29)
After
Quadratic ORIGINAL nonzero root29 square = 841
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S058
S060 · implies
Before
ORIGINAL nonzero root x=29 in1..66;actual square=841
After
Euclidean square=12*67+37;nonzero square class=37;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S059
S061 · calculation
Before
Quadratic ORIGINAL nonzero root30 square := mul(30, 30)
After
Quadratic ORIGINAL nonzero root30 square = 900
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S060
S062 · implies
Before
ORIGINAL nonzero root x=30 in1..66;actual square=900
After
Euclidean square=13*67+29;nonzero square class=29;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S061
S063 · calculation
Before
Quadratic ORIGINAL nonzero root31 square := mul(31, 31)
After
Quadratic ORIGINAL nonzero root31 square = 961
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S062
S064 · implies
Before
ORIGINAL nonzero root x=31 in1..66;actual square=961
After
Euclidean square=14*67+23;nonzero square class=23;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S063
S065 · calculation
Before
Quadratic ORIGINAL nonzero root32 square := mul(32, 32)
After
Quadratic ORIGINAL nonzero root32 square = 1024
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S064
S066 · implies
Before
ORIGINAL nonzero root x=32 in1..66;actual square=1024
After
Euclidean square=15*67+19;nonzero square class=19;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S065
S067 · calculation
Before
Quadratic ORIGINAL nonzero root33 square := mul(33, 33)
After
Quadratic ORIGINAL nonzero root33 square = 1089
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S066
S068 · implies
Before
ORIGINAL nonzero root x=33 in1..66;actual square=1089
After
Euclidean square=16*67+17;nonzero square class=17;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S067
S069 · calculation
Before
Quadratic ORIGINAL nonzero root34 square := mul(34, 34)
After
Quadratic ORIGINAL nonzero root34 square = 1156
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S068
S070 · implies
Before
ORIGINAL nonzero root x=34 in1..66;actual square=1156
After
Euclidean square=17*67+17;nonzero square class=17;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S069
S071 · calculation
Before
Quadratic ORIGINAL nonzero root35 square := mul(35, 35)
After
Quadratic ORIGINAL nonzero root35 square = 1225
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S070
S072 · implies
Before
ORIGINAL nonzero root x=35 in1..66;actual square=1225
After
Euclidean square=18*67+19;nonzero square class=19;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S071
S073 · calculation
Before
Quadratic ORIGINAL nonzero root36 square := mul(36, 36)
After
Quadratic ORIGINAL nonzero root36 square = 1296
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S072
S074 · implies
Before
ORIGINAL nonzero root x=36 in1..66;actual square=1296
After
Euclidean square=19*67+23;nonzero square class=23;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S073
S075 · calculation
Before
Quadratic ORIGINAL nonzero root37 square := mul(37, 37)
After
Quadratic ORIGINAL nonzero root37 square = 1369
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S074
S076 · implies
Before
ORIGINAL nonzero root x=37 in1..66;actual square=1369
After
Euclidean square=20*67+29;nonzero square class=29;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S075
S077 · calculation
Before
Quadratic ORIGINAL nonzero root38 square := mul(38, 38)
After
Quadratic ORIGINAL nonzero root38 square = 1444
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S076
S078 · implies
Before
ORIGINAL nonzero root x=38 in1..66;actual square=1444
After
Euclidean square=21*67+37;nonzero square class=37;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S077
S079 · calculation
Before
Quadratic ORIGINAL nonzero root39 square := mul(39, 39)
After
Quadratic ORIGINAL nonzero root39 square = 1521
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S078
S080 · implies
Before
ORIGINAL nonzero root x=39 in1..66;actual square=1521
After
Euclidean square=22*67+47;nonzero square class=47;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S079
S081 · calculation
Before
Quadratic ORIGINAL nonzero root40 square := mul(40, 40)
After
Quadratic ORIGINAL nonzero root40 square = 1600
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S080
S082 · implies
Before
ORIGINAL nonzero root x=40 in1..66;actual square=1600
After
Euclidean square=23*67+59;nonzero square class=59;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S081
S083 · calculation
Before
Quadratic ORIGINAL nonzero root41 square := mul(41, 41)
After
Quadratic ORIGINAL nonzero root41 square = 1681
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S082
S084 · implies
Before
ORIGINAL nonzero root x=41 in1..66;actual square=1681
After
Euclidean square=25*67+6;nonzero square class=6;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S083
S085 · calculation
Before
Quadratic ORIGINAL nonzero root42 square := mul(42, 42)
After
Quadratic ORIGINAL nonzero root42 square = 1764
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S084
S086 · implies
Before
ORIGINAL nonzero root x=42 in1..66;actual square=1764
After
Euclidean square=26*67+22;nonzero square class=22;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S085
S087 · calculation
Before
Quadratic ORIGINAL nonzero root43 square := mul(43, 43)
After
Quadratic ORIGINAL nonzero root43 square = 1849
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S086
S088 · implies
Before
ORIGINAL nonzero root x=43 in1..66;actual square=1849
After
Euclidean square=27*67+40;nonzero square class=40;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S087
S089 · calculation
Before
Quadratic ORIGINAL nonzero root44 square := mul(44, 44)
After
Quadratic ORIGINAL nonzero root44 square = 1936
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S088
S090 · implies
Before
ORIGINAL nonzero root x=44 in1..66;actual square=1936
After
Euclidean square=28*67+60;nonzero square class=60;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S089
S091 · calculation
Before
Quadratic ORIGINAL nonzero root45 square := mul(45, 45)
After
Quadratic ORIGINAL nonzero root45 square = 2025
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S090
S092 · implies
Before
ORIGINAL nonzero root x=45 in1..66;actual square=2025
After
Euclidean square=30*67+15;nonzero square class=15;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S091
S093 · calculation
Before
Quadratic ORIGINAL nonzero root46 square := mul(46, 46)
After
Quadratic ORIGINAL nonzero root46 square = 2116
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S092
S094 · implies
Before
ORIGINAL nonzero root x=46 in1..66;actual square=2116
After
Euclidean square=31*67+39;nonzero square class=39;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S093
S095 · calculation
Before
Quadratic ORIGINAL nonzero root47 square := mul(47, 47)
After
Quadratic ORIGINAL nonzero root47 square = 2209
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S094
S096 · implies
Before
ORIGINAL nonzero root x=47 in1..66;actual square=2209
After
Euclidean square=32*67+65;nonzero square class=65;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S095
S097 · calculation
Before
Quadratic ORIGINAL nonzero root48 square := mul(48, 48)
After
Quadratic ORIGINAL nonzero root48 square = 2304
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S096
S098 · implies
Before
ORIGINAL nonzero root x=48 in1..66;actual square=2304
After
Euclidean square=34*67+26;nonzero square class=26;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S097
S099 · calculation
Before
Quadratic ORIGINAL nonzero root49 square := mul(49, 49)
After
Quadratic ORIGINAL nonzero root49 square = 2401
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S098
S100 · implies
Before
ORIGINAL nonzero root x=49 in1..66;actual square=2401
After
Euclidean square=35*67+56;nonzero square class=56;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S099
S101 · calculation
Before
Quadratic ORIGINAL nonzero root50 square := mul(50, 50)
After
Quadratic ORIGINAL nonzero root50 square = 2500
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S100
S102 · implies
Before
ORIGINAL nonzero root x=50 in1..66;actual square=2500
After
Euclidean square=37*67+21;nonzero square class=21;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S101
S103 · calculation
Before
Quadratic ORIGINAL nonzero root51 square := mul(51, 51)
After
Quadratic ORIGINAL nonzero root51 square = 2601
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S102
S104 · implies
Before
ORIGINAL nonzero root x=51 in1..66;actual square=2601
After
Euclidean square=38*67+55;nonzero square class=55;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S103
S105 · calculation
Before
Quadratic ORIGINAL nonzero root52 square := mul(52, 52)
After
Quadratic ORIGINAL nonzero root52 square = 2704
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S104
S106 · implies
Before
ORIGINAL nonzero root x=52 in1..66;actual square=2704
After
Euclidean square=40*67+24;nonzero square class=24;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S105
S107 · calculation
Before
Quadratic ORIGINAL nonzero root53 square := mul(53, 53)
After
Quadratic ORIGINAL nonzero root53 square = 2809
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S106
S108 · implies
Before
ORIGINAL nonzero root x=53 in1..66;actual square=2809
After
Euclidean square=41*67+62;nonzero square class=62;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S107
S109 · calculation
Before
Quadratic ORIGINAL nonzero root54 square := mul(54, 54)
After
Quadratic ORIGINAL nonzero root54 square = 2916
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S108
S110 · implies
Before
ORIGINAL nonzero root x=54 in1..66;actual square=2916
After
Euclidean square=43*67+35;nonzero square class=35;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S109
S111 · calculation
Before
Quadratic ORIGINAL nonzero root55 square := mul(55, 55)
After
Quadratic ORIGINAL nonzero root55 square = 3025
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S110
S112 · implies
Before
ORIGINAL nonzero root x=55 in1..66;actual square=3025
After
Euclidean square=45*67+10;nonzero square class=10;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S111
S113 · calculation
Before
Quadratic ORIGINAL nonzero root56 square := mul(56, 56)
After
Quadratic ORIGINAL nonzero root56 square = 3136
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S112
S114 · implies
Before
ORIGINAL nonzero root x=56 in1..66;actual square=3136
After
Euclidean square=46*67+54;nonzero square class=54;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S113
S115 · calculation
Before
Quadratic ORIGINAL nonzero root57 square := mul(57, 57)
After
Quadratic ORIGINAL nonzero root57 square = 3249
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S114
S116 · implies
Before
ORIGINAL nonzero root x=57 in1..66;actual square=3249
After
Euclidean square=48*67+33;nonzero square class=33;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S115
S117 · calculation
Before
Quadratic ORIGINAL nonzero root58 square := mul(58, 58)
After
Quadratic ORIGINAL nonzero root58 square = 3364
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S116
S118 · implies
Before
ORIGINAL nonzero root x=58 in1..66;actual square=3364
After
Euclidean square=50*67+14;nonzero square class=14;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S117
S119 · calculation
Before
Quadratic ORIGINAL nonzero root59 square := mul(59, 59)
After
Quadratic ORIGINAL nonzero root59 square = 3481
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S118
S120 · implies
Before
ORIGINAL nonzero root x=59 in1..66;actual square=3481
After
Euclidean square=51*67+64;nonzero square class=64;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S119
S121 · calculation
Before
Quadratic ORIGINAL nonzero root60 square := mul(60, 60)
After
Quadratic ORIGINAL nonzero root60 square = 3600
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S120
S122 · implies
Before
ORIGINAL nonzero root x=60 in1..66;actual square=3600
After
Euclidean square=53*67+49;nonzero square class=49;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S121
S123 · calculation
Before
Quadratic ORIGINAL nonzero root61 square := mul(61, 61)
After
Quadratic ORIGINAL nonzero root61 square = 3721
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S122
S124 · implies
Before
ORIGINAL nonzero root x=61 in1..66;actual square=3721
After
Euclidean square=55*67+36;nonzero square class=36;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S123
S125 · calculation
Before
Quadratic ORIGINAL nonzero root62 square := mul(62, 62)
After
Quadratic ORIGINAL nonzero root62 square = 3844
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S124
S126 · implies
Before
ORIGINAL nonzero root x=62 in1..66;actual square=3844
After
Euclidean square=57*67+25;nonzero square class=25;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S125
S127 · calculation
Before
Quadratic ORIGINAL nonzero root63 square := mul(63, 63)
After
Quadratic ORIGINAL nonzero root63 square = 3969
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S126
S128 · implies
Before
ORIGINAL nonzero root x=63 in1..66;actual square=3969
After
Euclidean square=59*67+16;nonzero square class=16;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S127
S129 · calculation
Before
Quadratic ORIGINAL nonzero root64 square := mul(64, 64)
After
Quadratic ORIGINAL nonzero root64 square = 4096
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S128
S130 · implies
Before
ORIGINAL nonzero root x=64 in1..66;actual square=4096
After
Euclidean square=61*67+9;nonzero square class=9;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S129
S131 · calculation
Before
Quadratic ORIGINAL nonzero root65 square := mul(65, 65)
After
Quadratic ORIGINAL nonzero root65 square = 4225
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S130
S132 · implies
Before
ORIGINAL nonzero root x=65 in1..66;actual square=4225
After
Euclidean square=63*67+4;nonzero square class=4;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S131
S133 · calculation
Before
Quadratic ORIGINAL nonzero root66 square := mul(66, 66)
After
Quadratic ORIGINAL nonzero root66 square = 4356
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S132
S134 · implies
Before
ORIGINAL nonzero root x=66 in1..66;actual square=4356
After
Euclidean square=65*67+1;nonzero square class=1;complete root-index order retained
Conditions: none
Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.
Step identifiers and dependencies
Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S133
S135 · calculation
Before
Quadratic distinct nonzero residue count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
After
Quadratic distinct nonzero residue count = 33
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S134
S136 · calculation
Before
Quadratic ORIGINAL modulus minus1 := sub(67, 1)
After
Quadratic ORIGINAL modulus minus1 = 66
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S135
S137 · calculation
Before
Quadratic half of all nonzero roots := div(66, 2)
After
Quadratic half of all nonzero roots = 33
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S136
S138 · implies
Before
ORIGINAL ODD PRIME p=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]];distinct 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
x^2=y^2 implies p divides(x-y)(x+y),so x=y or x=-y modp;odd p and nonzero x make opposite roots DISTINCT;each image has2 roots,so residue count=33=(67-1)/2=33
Conditions: none
The prime-field no-zero-divisor property gives x=±y;oddness makes2x nonzero.The opposite pairs partition all p−1 nonzero roots.This proves both counting and two-root statements,not their extension to zero,2 or composite moduli.
Step identifiers and dependencies
Rule: quadratic_prime_field_opposite_root_pairs_exactly_two · Uses: G001, S137
S139 · calculation
Before
Quadratic ORIGINAL candidate class Euclidean quotient times modulus := mul(0, 67)
After
Quadratic ORIGINAL candidate class Euclidean quotient times modulus = 0
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S138
S140 · calculation
Before
Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(56, 0)
After
Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple = 56
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S139
S141 · implies
Before
Quadratic ORIGINAL candidate class:ORIGINAL signed value=56;positive modulus=67;actual quotient=0
After
canonical residue=56;value=quotient*modulus+residue and0<=residue<modulus
Conditions: none
Retain the original signed value and congruence class via Euclidean division;do not mistake a language-specific negative remainder for the canonical class.
Step identifiers and dependencies
Rule: signed_mod_ORIGINAL_class_normalized_not_changed · Uses: G001, S140
S142 · calculation
Before
Quadratic negative of FIRST source root Euclidean quotient times modulus := mul(-1, 67)
After
Quadratic negative of FIRST source root Euclidean quotient times modulus = -67
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S141
S143 · calculation
Before
Quadratic negative of FIRST source root ORIGINAL signed value minus multiple := sub(-18, -67)
After
Quadratic negative of FIRST source root ORIGINAL signed value minus multiple = 49
Conditions: none
Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S142
S144 · implies
Before
Quadratic negative of FIRST source root:ORIGINAL signed value=-18;positive modulus=67;actual quotient=-1
After
canonical residue=49;value=quotient*modulus+residue and0<=residue<modulus
Conditions: none
Retain the original signed value and congruence class via Euclidean division;do not mistake a language-specific negative remainder for the canonical class.
Step identifiers and dependencies
Rule: signed_mod_ORIGINAL_class_normalized_not_changed · Uses: G001, S143