# Question 23367: A Nonzero Residue Modulo an Odd Prime Has Two Distinct Opposite Roots [Return to catalogue page 117](https://ocme.evemisslab.com/en/questions/page/117/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r06-quadratic-residue-foundations/#q-20261014-advance-20261111-r06-quadratic-residue-0067) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0045-23367 · q-20261014-advance-20261111-r06-quadratic-residue-0067 ## Original question and givens Use bounded odd prime p=61 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Original a=45. Give (p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, certificate). - G001 — Use bounded odd prime p=61 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Original a=45. Give (p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-quadratic-residue](https://ocme.evemisslab.com/en/mko/mko-quadratic-residue/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore Quadratic ORIGINAL modulus:ORIGINAL integer=61;complete possible trial divisors=[2, 3, 4, 5, 6, 7] After ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 5]];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 dependenciesRule: finite_prime_ALL_trial_divisors_through_square_root · Uses: G001, S001 S003 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore ORIGINAL nonzero root x=1 in1..60;actual square=1 After Euclidean square=0*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S003 S005 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore ORIGINAL nonzero root x=2 in1..60;actual square=4 After Euclidean square=0*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S005 S007 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL nonzero root x=3 in1..60;actual square=9 After Euclidean square=0*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S007 S009 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore ORIGINAL nonzero root x=4 in1..60;actual square=16 After Euclidean square=0*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S009 S011 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore ORIGINAL nonzero root x=5 in1..60;actual square=25 After Euclidean square=0*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S011 S013 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore ORIGINAL nonzero root x=6 in1..60;actual square=36 After Euclidean square=0*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S013 S015 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore ORIGINAL nonzero root x=7 in1..60;actual square=49 After Euclidean square=0*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S015 S017 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore ORIGINAL nonzero root x=8 in1..60;actual square=64 After Euclidean square=1*61+3;nonzero square class=3;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S017 S019 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore ORIGINAL nonzero root x=9 in1..60;actual square=81 After Euclidean square=1*61+20;nonzero square class=20;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S019 S021 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore ORIGINAL nonzero root x=10 in1..60;actual square=100 After Euclidean square=1*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S021 S023 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore ORIGINAL nonzero root x=11 in1..60;actual square=121 After Euclidean square=1*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S023 S025 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore ORIGINAL nonzero root x=12 in1..60;actual square=144 After Euclidean square=2*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S025 S027 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore ORIGINAL nonzero root x=13 in1..60;actual square=169 After Euclidean square=2*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S027 S029 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore ORIGINAL nonzero root x=14 in1..60;actual square=196 After Euclidean square=3*61+13;nonzero square class=13;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S029 S031 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore ORIGINAL nonzero root x=15 in1..60;actual square=225 After Euclidean square=3*61+42;nonzero square class=42;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S031 S033 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore ORIGINAL nonzero root x=16 in1..60;actual square=256 After Euclidean square=4*61+12;nonzero square class=12;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S033 S035 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · impliesBefore ORIGINAL nonzero root x=17 in1..60;actual square=289 After Euclidean square=4*61+45;nonzero square class=45;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S035 S037 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore ORIGINAL nonzero root x=18 in1..60;actual square=324 After Euclidean square=5*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S037 S039 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · impliesBefore ORIGINAL nonzero root x=19 in1..60;actual square=361 After Euclidean square=5*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S039 S041 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore ORIGINAL nonzero root x=20 in1..60;actual square=400 After Euclidean square=6*61+34;nonzero square class=34;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S041 S043 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore ORIGINAL nonzero root x=21 in1..60;actual square=441 After Euclidean square=7*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S043 S045 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · impliesBefore ORIGINAL nonzero root x=22 in1..60;actual square=484 After Euclidean square=7*61+57;nonzero square class=57;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S045 S047 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · impliesBefore ORIGINAL nonzero root x=23 in1..60;actual square=529 After Euclidean square=8*61+41;nonzero square class=41;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S047 S049 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · impliesBefore ORIGINAL nonzero root x=24 in1..60;actual square=576 After Euclidean square=9*61+27;nonzero square class=27;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S049 S051 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · impliesBefore ORIGINAL nonzero root x=25 in1..60;actual square=625 After Euclidean square=10*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S051 S053 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · impliesBefore ORIGINAL nonzero root x=26 in1..60;actual square=676 After Euclidean square=11*61+5;nonzero square class=5;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S053 S055 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · impliesBefore ORIGINAL nonzero root x=27 in1..60;actual square=729 After Euclidean square=11*61+58;nonzero square class=58;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S055 S057 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · impliesBefore ORIGINAL nonzero root x=28 in1..60;actual square=784 After Euclidean square=12*61+52;nonzero square class=52;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S057 S059 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · impliesBefore ORIGINAL nonzero root x=29 in1..60;actual square=841 After Euclidean square=13*61+48;nonzero square class=48;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S059 S061 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · impliesBefore ORIGINAL nonzero root x=30 in1..60;actual square=900 After Euclidean square=14*61+46;nonzero square class=46;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S061 S063 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · impliesBefore ORIGINAL nonzero root x=31 in1..60;actual square=961 After Euclidean square=15*61+46;nonzero square class=46;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S063 S065 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · impliesBefore ORIGINAL nonzero root x=32 in1..60;actual square=1024 After Euclidean square=16*61+48;nonzero square class=48;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S065 S067 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · impliesBefore ORIGINAL nonzero root x=33 in1..60;actual square=1089 After Euclidean square=17*61+52;nonzero square class=52;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S067 S069 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · impliesBefore ORIGINAL nonzero root x=34 in1..60;actual square=1156 After Euclidean square=18*61+58;nonzero square class=58;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S069 S071 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · impliesBefore ORIGINAL nonzero root x=35 in1..60;actual square=1225 After Euclidean square=20*61+5;nonzero square class=5;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S071 S073 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · impliesBefore ORIGINAL nonzero root x=36 in1..60;actual square=1296 After Euclidean square=21*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S073 S075 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · impliesBefore ORIGINAL nonzero root x=37 in1..60;actual square=1369 After Euclidean square=22*61+27;nonzero square class=27;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S075 S077 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · impliesBefore ORIGINAL nonzero root x=38 in1..60;actual square=1444 After Euclidean square=23*61+41;nonzero square class=41;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S077 S079 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · impliesBefore ORIGINAL nonzero root x=39 in1..60;actual square=1521 After Euclidean square=24*61+57;nonzero square class=57;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S079 S081 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · impliesBefore ORIGINAL nonzero root x=40 in1..60;actual square=1600 After Euclidean square=26*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S081 S083 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · impliesBefore ORIGINAL nonzero root x=41 in1..60;actual square=1681 After Euclidean square=27*61+34;nonzero square class=34;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S083 S085 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · impliesBefore ORIGINAL nonzero root x=42 in1..60;actual square=1764 After Euclidean square=28*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S085 S087 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · impliesBefore ORIGINAL nonzero root x=43 in1..60;actual square=1849 After Euclidean square=30*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S087 S089 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · impliesBefore ORIGINAL nonzero root x=44 in1..60;actual square=1936 After Euclidean square=31*61+45;nonzero square class=45;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S089 S091 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S090 S092 · impliesBefore ORIGINAL nonzero root x=45 in1..60;actual square=2025 After Euclidean square=33*61+12;nonzero square class=12;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S091 S093 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S092 S094 · impliesBefore ORIGINAL nonzero root x=46 in1..60;actual square=2116 After Euclidean square=34*61+42;nonzero square class=42;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S093 S095 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S094 S096 · impliesBefore ORIGINAL nonzero root x=47 in1..60;actual square=2209 After Euclidean square=36*61+13;nonzero square class=13;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S095 S097 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · impliesBefore ORIGINAL nonzero root x=48 in1..60;actual square=2304 After Euclidean square=37*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S097 S099 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S098 S100 · impliesBefore ORIGINAL nonzero root x=49 in1..60;actual square=2401 After Euclidean square=39*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S099 S101 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S100 S102 · impliesBefore ORIGINAL nonzero root x=50 in1..60;actual square=2500 After Euclidean square=40*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S101 S103 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · impliesBefore ORIGINAL nonzero root x=51 in1..60;actual square=2601 After Euclidean square=42*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S103 S105 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · impliesBefore ORIGINAL nonzero root x=52 in1..60;actual square=2704 After Euclidean square=44*61+20;nonzero square class=20;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S105 S107 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · impliesBefore ORIGINAL nonzero root x=53 in1..60;actual square=2809 After Euclidean square=46*61+3;nonzero square class=3;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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S107 S109 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S108 S110 · impliesBefore ORIGINAL nonzero root x=54 in1..60;actual square=2916 After Euclidean square=47*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S109 S111 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S110 S112 · impliesBefore ORIGINAL nonzero root x=55 in1..60;actual square=3025 After Euclidean square=49*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S111 S113 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S112 S114 · impliesBefore ORIGINAL nonzero root x=56 in1..60;actual square=3136 After Euclidean square=51*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S113 S115 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S114 S116 · impliesBefore ORIGINAL nonzero root x=57 in1..60;actual square=3249 After Euclidean square=53*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S115 S117 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S116 S118 · impliesBefore ORIGINAL nonzero root x=58 in1..60;actual square=3364 After Euclidean square=55*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S117 S119 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S118 S120 · impliesBefore ORIGINAL nonzero root x=59 in1..60;actual square=3481 After Euclidean square=57*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S119 S121 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S120 S122 · impliesBefore ORIGINAL nonzero root x=60 in1..60;actual square=3600 After Euclidean square=59*61+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 dependenciesRule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S121 S123 · calculationBefore 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) After Quadratic distinct nonzero residue count = 30 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S122 S124 · calculationBefore Quadratic ORIGINAL modulus minus1 := sub(61, 1) After Quadratic ORIGINAL modulus minus1 = 60 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123 S125 · calculationBefore Quadratic half of all nonzero roots := div(60, 2) After Quadratic half of all nonzero roots = 30 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124 S126 · impliesBefore ORIGINAL ODD PRIME p=61;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 3], [9, 20], [10, 39], [11, 60], [12, 22], [13, 47], [14, 13], [15, 42], [16, 12], [17, 45], [18, 19], [19, 56], [20, 34], [21, 14], [22, 57], [23, 41], [24, 27], [25, 15], [26, 5], [27, 58], [28, 52], [29, 48], [30, 46], [31, 46], [32, 48], [33, 52], [34, 58], [35, 5], [36, 15], [37, 27], [38, 41], [39, 57], [40, 14], [41, 34], [42, 56], [43, 19], [44, 45], [45, 12], [46, 42], [47, 13], [48, 47], [49, 22], [50, 60], [51, 39], [52, 20], [53, 3], [54, 49], [55, 36], [56, 25], [57, 16], [58, 9], [59, 4], [60, 1]];distinct classes=[1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 52, 56, 57, 58, 60] 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=30=(61-1)/2=30 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 dependenciesRule: quadratic_prime_field_opposite_root_pairs_exactly_two · Uses: G001, S125 S127 · calculationBefore Quadratic ORIGINAL candidate class Euclidean quotient times modulus := mul(0, 61) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S126 S128 · calculationBefore Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(45, 0) After Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple = 45 Conditions: none Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127 S129 · impliesBefore Quadratic ORIGINAL candidate class:ORIGINAL signed value=45;positive modulus=61;actual quotient=0 After canonical residue=45;value=quotient*modulus+residue and0<=residue