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Question 23353: A Nonzero Residue Modulo an Odd Prime Has Two Distinct Opposite Roots

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0045-23353 · q-20261014-advance-20261111-r06-quadratic-residue-0053

Original question and givens

Use bounded odd prime p=7 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=7 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.

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=7;complete possible trial divisors=[2]

After

ALL [divisor,remainder] rows=[[2, 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 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..6;actual square=1

After

Euclidean square=0*7+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..6;actual square=4

After

Euclidean square=0*7+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..6;actual square=9

After

Euclidean square=1*7+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 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..6;actual square=16

After

Euclidean square=2*7+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 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..6;actual square=25

After

Euclidean square=3*7+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, 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..6;actual square=36

After

Euclidean square=5*7+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, S013

S015 · calculation

Before

Quadratic distinct nonzero residue count := sum(1, 1, 1)

After

Quadratic distinct nonzero residue count = 3

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 · calculation

Before

Quadratic ORIGINAL modulus minus1 := sub(7, 1)

After

Quadratic ORIGINAL modulus minus1 = 6

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, S015

S017 · calculation

Before

Quadratic half of all nonzero roots := div(6, 2)

After

Quadratic half of all nonzero roots = 3

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 ODD PRIME p=7;complete nonzero square table=[[1, 1], [2, 4], [3, 2], [4, 2], [5, 4], [6, 1]];distinct classes=[1, 2, 4]

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=3=(7-1)/2=3

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, S017

S019 · calculation

Before

Quadratic ORIGINAL candidate class Euclidean quotient times modulus := mul(0, 7)

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, S018

S020 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

Quadratic ORIGINAL candidate class:ORIGINAL signed value=2;positive modulus=7;actual quotient=0

After

canonical residue=2;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, S020

S022 · calculation

Before

Quadratic negative of FIRST source root Euclidean quotient times modulus := mul(-1, 7)

After

Quadratic negative of FIRST source root Euclidean quotient times modulus = -7

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, S021

S023 · calculation

Before

Quadratic negative of FIRST source root ORIGINAL signed value minus multiple := sub(-3, -7)

After

Quadratic negative of FIRST source root ORIGINAL signed value minus multiple = 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, S022

S024 · implies

Before

Quadratic negative of FIRST source root:ORIGINAL signed value=-3;positive modulus=7;actual quotient=-1

After

canonical residue=4;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, S023

Reverse checks

  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Recompute Quadratic negative of FIRST source root ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Quadratic negative of FIRST source root Euclidean quotient times modulus with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Recompute Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Quadratic ORIGINAL candidate class Euclidean quotient times modulus with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Recompute Quadratic half of all nonzero roots with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Quadratic ORIGINAL modulus minus1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Quadratic distinct nonzero residue count with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root6 square with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root5 square with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root4 square with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root3 square with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root2 square with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root1 square with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(7, 2, [3, 4], 4, Yes, Yes, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: e46bf80d3e19886ca972f4de26f523e53b0993da0c40808f44ce996fd27ec7d5
  • candidate record SHA-256: 83ffa1874518c71c5c53a6fd21d820945286aff8c4b8515b0078ff5144d1e2ae
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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