# Question 35643: Counterexample to the Pythagorean Relation [Return to catalogue page 179](https://ocme.evemisslab.com/en/questions/page/179/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260912-geometry-foundations/#q-20260912-pythagorean-counterexample-0008) 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. R0070-35643 · q-20260912-pythagorean-counterexample-0008 ## Original question and givens Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c? A: 11, 60, 61; B: 11, 60, 61; C: 11, 60, 61; D: 11, 61, 61 - G001 — Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c? A: 11, 60, 61; B: 11, 60, 61; C: 11, 60, 61; D: 11, 61, 61 ## 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-euclid-pythagorean-theorem](https://ocme.evemisslab.com/en/mko/mko-euclid-pythagorean-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-pythagorean-counterexample, [counterexample_index, mode, options]] After derived = [counter, null, null] Conditions: none Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules. Step identifiers and dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001 S002 · impliesBefore source roles = [[[11, 60, 61], [11, 60, 61], [11, 60, 61], [11, 61, 61]]] After derived = [[0: 甲 = A, 1: 乙 = B, 2: 丙 = C, 3: 丁 = D]] Conditions: none Keep all FOUR options in ORIGINAL positions,even repeated valid triples.Chinese 甲乙丙丁 corresponds POSITIONALLY to English ABCD.The LAST coordinate is c;do not sort and silently change that role. Step identifiers and dependenciesRule: Pythagorean_OPTION_position_legend_keep_duplicate_options · Uses: G001, S001 S003 · calculationBefore Pyth option 0 a squared := mul(11, 11) After Pyth option 0 a squared = 121 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Pyth option 0 b squared := mul(60, 60) After Pyth option 0 b squared = 3600 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Pyth option 0 LAST c squared := mul(61, 61) After Pyth option 0 LAST c squared = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Pyth option 0 sum a2 b2 := sum(121, 3600) After Pyth option 0 sum a2 b2 = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [0, [11, 60, 61]] After derived = [Yes, 3721, 3721, Yes] Conditions: none For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse. Step identifiers and dependenciesRule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S006 S008 · calculationBefore Pyth option 1 a squared := mul(11, 11) After Pyth option 1 a squared = 121 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Pyth option 1 b squared := mul(60, 60) After Pyth option 1 b squared = 3600 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Pyth option 1 LAST c squared := mul(61, 61) After Pyth option 1 LAST c squared = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Pyth option 1 sum a2 b2 := sum(121, 3600) After Pyth option 1 sum a2 b2 = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore source roles = [1, [11, 60, 61]] After derived = [Yes, 3721, 3721, Yes] Conditions: none For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse. Step identifiers and dependenciesRule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S011 S013 · calculationBefore Pyth option 2 a squared := mul(11, 11) After Pyth option 2 a squared = 121 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Pyth option 2 b squared := mul(60, 60) After Pyth option 2 b squared = 3600 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Pyth option 2 LAST c squared := mul(61, 61) After Pyth option 2 LAST c squared = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Pyth option 2 sum a2 b2 := sum(121, 3600) After Pyth option 2 sum a2 b2 = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [2, [11, 60, 61]] After derived = [Yes, 3721, 3721, Yes] Conditions: none For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse. Step identifiers and dependenciesRule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S016 S018 · calculationBefore Pyth option 3 a squared := mul(11, 11) After Pyth option 3 a squared = 121 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Pyth option 3 b squared := mul(61, 61) After Pyth option 3 b squared = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Pyth option 3 LAST c squared := mul(61, 61) After Pyth option 3 LAST c squared = 3721 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Pyth option 3 sum a2 b2 := sum(121, 3721) After Pyth option 3 sum a2 b2 = 3842 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore source roles = [3, [11, 61, 61]] After derived = [Yes, 3842, 3721, No] Conditions: none For EACH option,check positive entries AND a²+b²=c².This directly checks the STATED squared relation,not a drawing/largest-sort inference or an unneeded geometric converse. Step identifiers and dependenciesRule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S021 S023 · impliesBefore source roles = [[Yes, Yes, Yes, No], [3]] After derived = [丁] Conditions: none Select the ORIGINAL label ONLY after ALL FOUR checks establish exactly one failure.Check counterexample_index metadata LAST;it never substitutes for testing the options. Step identifiers and dependenciesRule: Pythagorean_UNIQUE_invalid_after_ALL_four_checks · Uses: G001, S022 ## Reverse checks - S023 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S022 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S021 · premise_and_result_audit — Recompute Pyth option 3 sum a2 b2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Pyth option 3 LAST c squared with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Pyth option 3 b squared with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Pyth option 3 a squared with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S016 · premise_and_result_audit — Recompute Pyth option 2 sum a2 b2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Pyth option 2 LAST c squared with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Pyth option 2 b squared with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Pyth option 2 a squared with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S011 · premise_and_result_audit — Recompute Pyth option 1 sum a2 b2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Pyth option 1 LAST c squared with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Pyth option 1 b squared with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Pyth option 1 a squared with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S006 · premise_and_result_audit — Recompute Pyth option 0 sum a2 b2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Pyth option 0 LAST c squared with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Pyth option 0 b squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Pyth option 0 a squared with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. ## Result D 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: 635813d18b2c365aacce34292745696cea62a270fe78d6e70000745ca182a47d - candidate record SHA-256: eb36bd830ce23ce15b4c8caa3fd753fbf554a4e982c2008b908bf15ffa445627 - semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80 - question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260912-pythagorean-counterexample-0008.json)