Proof portfolio: from a remainder procedure to a theorem
Build one coherent mathematical manuscript over several sessions. Investigate which integers can be written as ax+by, prove a terminating procedure for finding their positive generator, and use the result to classify every solution of a linear integer equation. Examples motivate claims; they do not prove universal statements. Keep conjectures, rejected arguments, revisions and reviewer feedback. Public guidance is available, so this is a supported portfolio rather than an unseen examination.
Planning estimate: 12 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Preparation and working agreement
Use integers throughout. An integer d divides n when n=dk for some integer k. For integers a,b not both zero, gcd(a,b) means their greatest positive common divisor. We do not define gcd(0,0) in this investigation.
You may use the division algorithm: if A is a nonnegative integer and B a positive integer, there are unique integers q,r with A=qB+r and 0≤r<B. You may also use the well-ordering principle for nonempty sets of nonnegative integers. Identify every further theorem you use and prove it or cite an established lesson argument.
Begin with a=84,b=30, then test a negative input, one zero input, equal inputs and relatively prime inputs. A table or short program can check examples, but your main deliverable is a proof. Record exact arithmetic and enough intermediate steps for another person to reproduce the checks.
Recommended route: establish the general result for all integer a,b not both zero. Supported first-pass route: prove it for positive a,b, then handle signs and a zero input in a separate extension. Clearly label any unproved extension rather than claiming the general theorem.
Keep a dated source and assistance ledger, including revealed guidance. Plan an initial conjecture, proof draft, adversarial review, revised manuscript and short defense. A peer or teacher must actually review the argument before you report independent review. The suggested hours are an author planning estimate, not measured learner time.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
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0 of 6 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
State a conjecture and its logical boundaries
For fixed integers a,b not both zero, define S={ax+by : x,y are integers}. Explore S for (84,30), (−84,30), (0,30) and (7,5). Conjecture a description using g=gcd(a,b). State the proposed set equality as two quantified implications. Write its negation with an explicit witness condition. Distinguish “for every c there exist x,y” from choosing one pair x,y for all c. List what your examples establish and what remains to prove.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove the invariant and termination
Normalize signs to A=|a| and B=|b|. If B>0, replace (A,B) with (B,r), where A=qB+r and 0≤r<B; stop when the second coordinate is zero. Prove that each replacement preserves the full set of integer linear combinations, by constructing coefficient witnesses in both directions. Deduce preservation of positive common divisors. Prove termination using a decreasing nonnegative integer and identify the final first coordinate as g. Include an initial zero coordinate and explain why division by the second coordinate is performed only when it is positive.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Recover a witness and prove the set equality
Trace the procedure for (84,30), recover integers u,v with 84u+30v=g, and verify your arithmetic independently. Then prove that a terminal witness can always be transported backward to obtain au+bv=g for general admissible inputs. Complete both inclusions S⊆gZ and gZ⊆S, including negative multiples and zero. Explain which part of your argument establishes existence and which part supplies a constructive method.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Classify all solutions and prove uniqueness of the parameter
For a,b not both zero, show that ax+by=c has integer solutions exactly when g divides c. Given one solution (x0,y0), prove that every solution is (x0+(b/g)t, y0−(a/g)t) for a unique integer t. First treat a and b both nonzero, proving any coprime-divisibility lemma you need from your witness theorem. Then handle a=0 or b=0. Apply the result to 84x+30y=18 and give a complete parametrization with an explicit inverse formula for t.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Audit failed proofs and revise the manuscript
Audit three proposed shortcuts: (i) “the remainder gets smaller, so this procedure terminates for positive real inputs too”; (ii) “g divides every combination, so g itself must be a combination”; (iii) “one parametrized family satisfies the equation, so it contains every solution.” Identify the exact missing reasoning and supply a counterexample or repair. For (i), distinguish integer descent from real descent, and either exhibit or rigorously explain a nonterminating real-input case. Incorporate the repairs into a complete manuscript, preserving an earlier revision and a change log tied to specific feedback. State whether feedback came from self-review, a peer or a teacher.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Submit and defend one coherent proof portfolio
Assemble a readable manuscript containing definitions, the main theorem, prerequisite lemmas, termination and invariant proofs, witness construction, complete solution classification, boundary cases and the source/assistance ledger. Include one fully checked example not used above. Ask a reviewer to choose a step for oral or written defense and a fresh equation to classify; record the question, your unaided response, subsequent corrections and the exact manuscript revision reviewed. If no reviewer is available, record a self-defense and leave independent review pending. Explain the distinction between submitting all milestones and demonstrating a correct proof.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
