# Question 23356: 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-0056) 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. R0045-23356 · q-20261014-advance-20261111-r06-quadratic-residue-0056 ## Original question and givens Use bounded odd prime p=17 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=2. Give (p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, certificate). - G001 — Use bounded odd prime p=17 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=2. 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=17;complete possible trial divisors=[2, 3, 4] After ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 1]];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..16;actual square=1 After Euclidean square=0*17+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..16;actual square=4 After Euclidean square=0*17+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..16;actual square=9 After Euclidean square=0*17+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..16;actual square=16 After Euclidean square=0*17+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..16;actual square=25 After Euclidean square=1*17+8;nonzero square class=8;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..16;actual square=36 After Euclidean square=2*17+2;nonzero square class=2;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..16;actual square=49 After Euclidean square=2*17+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, 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..16;actual square=64 After Euclidean square=3*17+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, 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..16;actual square=81 After Euclidean square=4*17+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, 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..16;actual square=100 After Euclidean square=5*17+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, 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..16;actual square=121 After Euclidean square=7*17+2;nonzero square class=2;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..16;actual square=144 After Euclidean square=8*17+8;nonzero square class=8;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..16;actual square=169 After Euclidean square=9*17+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, 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..16;actual square=196 After Euclidean square=11*17+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, 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..16;actual square=225 After Euclidean square=13*17+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, 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..16;actual square=256 After Euclidean square=15*17+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, S033 S035 · calculationBefore Quadratic distinct nonzero residue count := sum(1, 1, 1, 1, 1, 1, 1, 1) After Quadratic distinct nonzero residue count = 8 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 · calculationBefore Quadratic ORIGINAL modulus minus1 := sub(17, 1) After Quadratic ORIGINAL modulus minus1 = 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, S035 S037 · calculationBefore Quadratic half of all nonzero roots := div(16, 2) After Quadratic half of all nonzero roots = 8 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 ODD PRIME p=17;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 8], [6, 2], [7, 15], [8, 13], [9, 13], [10, 15], [11, 2], [12, 8], [13, 16], [14, 9], [15, 4], [16, 1]];distinct classes=[1, 2, 4, 8, 9, 13, 15, 16] 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=8=(17-1)/2=8 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, S037 S039 · calculationBefore Quadratic ORIGINAL candidate class Euclidean quotient times modulus := mul(0, 17) 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, S038 S040 · calculationBefore Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(2, 0) After Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple = 2 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, S039 S041 · impliesBefore Quadratic ORIGINAL candidate class:ORIGINAL signed value=2;positive modulus=17;actual quotient=0 After canonical residue=2;value=quotient*modulus+residue and0<=residue