# Question 23376: The Complete Nonzero Square Table Has Half as Many Distinct Residues [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-0001) 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. R0045-23376 · q-20261014-advance-20261111-r06-quadratic-residue-0001 ## Original question and givens Use bounded odd prime p=3 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. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal). - G001 — Use bounded odd prime p=3 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. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal). ## 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-square-table;original snapshot and mathematics retained After explicit ordered source output fields=[p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal] 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=3;complete possible trial divisors=[] After ALL [divisor,remainder] rows=[];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..2;actual square=1 After Euclidean square=0*3+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..2;actual square=4 After Euclidean square=1*3+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, S005 S007 · calculationBefore Quadratic distinct nonzero residue count := sum(1) After Quadratic distinct nonzero residue count = 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, S006 S008 · calculationBefore Quadratic ORIGINAL modulus minus1 := sub(3, 1) After Quadratic ORIGINAL modulus minus1 = 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, S007 S009 · calculationBefore Quadratic half of all nonzero roots := div(2, 2) After Quadratic half of all nonzero roots = 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, S008 S010 · impliesBefore ORIGINAL ODD PRIME p=3;complete nonzero square table=[[1, 1], [2, 1]];distinct classes=[1] 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=1=(3-1)/2=1 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, S009 ## Reverse checks - 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 half of all nonzero roots with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Quadratic ORIGINAL modulus minus1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Quadratic distinct nonzero residue count 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 (3, [[1, 1], [2, 1]], [1], 1, 1, Yes) 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: 627cbf63493ee9549aea9c97291b71f5cfe517109d05640b3776d622db894c43 - candidate record SHA-256: 214cf3673086ba97cde2ad3d3b97ca6ba9efa60cb4ddec4771ee812fc016311f - semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438 - question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r06-quadratic-residue-0001.json)