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Question 35646: Counterexample to the Pythagorean Relation

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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R0070-35646 · q-20260912-pythagorean-counterexample-0011

Original question and givens

Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c? A: 16, 63, 65; B: 16, 63, 65; C: 16, 64, 65; D: 16, 63, 65

  • G001 — Which positive triple does not satisfy a²+b²=c², taking the last number in each triple as c? A: 16, 63, 65; B: 16, 63, 65; C: 16, 64, 65; D: 16, 63, 65

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 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 dependencies

Rule: R70_SOURCE_bank_specific_interface · Uses: G001

S002 · implies

Before

source roles = [[[16, 63, 65], [16, 63, 65], [16, 64, 65], [16, 63, 65]]]

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 dependencies

Rule: Pythagorean_OPTION_position_legend_keep_duplicate_options · Uses: G001, S001

S003 · calculation

Before

Pyth option 0 a squared := mul(16, 16)

After

Pyth option 0 a squared = 256

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Pyth option 0 b squared := mul(63, 63)

After

Pyth option 0 b squared = 3969

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Pyth option 0 LAST c squared := mul(65, 65)

After

Pyth option 0 LAST c squared = 4225

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Pyth option 0 sum a2 b2 := sum(256, 3969)

After

Pyth option 0 sum a2 b2 = 4225

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [0, [16, 63, 65]]

After

derived = [Yes, 4225, 4225, 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 dependencies

Rule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S006

S008 · calculation

Before

Pyth option 1 a squared := mul(16, 16)

After

Pyth option 1 a squared = 256

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Pyth option 1 b squared := mul(63, 63)

After

Pyth option 1 b squared = 3969

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Pyth option 1 LAST c squared := mul(65, 65)

After

Pyth option 1 LAST c squared = 4225

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Pyth option 1 sum a2 b2 := sum(256, 3969)

After

Pyth option 1 sum a2 b2 = 4225

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [1, [16, 63, 65]]

After

derived = [Yes, 4225, 4225, 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 dependencies

Rule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S011

S013 · calculation

Before

Pyth option 2 a squared := mul(16, 16)

After

Pyth option 2 a squared = 256

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Pyth option 2 b squared := mul(64, 64)

After

Pyth option 2 b squared = 4096

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Pyth option 2 LAST c squared := mul(65, 65)

After

Pyth option 2 LAST c squared = 4225

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Pyth option 2 sum a2 b2 := sum(256, 4096)

After

Pyth option 2 sum a2 b2 = 4352

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [2, [16, 64, 65]]

After

derived = [Yes, 4352, 4225, 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 dependencies

Rule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S016

S018 · calculation

Before

Pyth option 3 a squared := mul(16, 16)

After

Pyth option 3 a squared = 256

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Pyth option 3 b squared := mul(63, 63)

After

Pyth option 3 b squared = 3969

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Pyth option 3 LAST c squared := mul(65, 65)

After

Pyth option 3 LAST c squared = 4225

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Pyth option 3 sum a2 b2 := sum(256, 3969)

After

Pyth option 3 sum a2 b2 = 4225

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [3, [16, 63, 65]]

After

derived = [Yes, 4225, 4225, 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 dependencies

Rule: Pythagorean_EACH_positive_triple_square_relation · Uses: G001, S021

S023 · implies

Before

source roles = [[Yes, Yes, No, Yes], [2]]

After

derived = [丙]

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 dependencies

Rule: 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

C

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: 934152a73b2136f4dc13fbb8af9c19e3363dc24d41034e020b1e54c6fb8c07ad
  • candidate record SHA-256: f53dc113a1280ec30d19edb0bc36c83d8e5787ca3e503ed03d38d54e19c5edfc
  • semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
  • question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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