Pipeline-math (2026): Erdős problem 477

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corollary_1_5: An integer outside the thirteenth powers gives a diagonal affine surface with no nonconstant rational one-parameter curve over the rationals.

evidence/: Source-owned compilation review, coverage and distinct grade, with exact premises and an accepted native transformation.

lemma_1_4: A rational parametrized curve on the diagonal thirteenth-power surface forces the constant to be a rational thirteenth power and lies on one of three explicit lines.

lemma_1_7: Finite avoidance of a difference set allows a sequence of disjoint translates that eventually covers every integer.

proposition_1_6: For each fixed integer outside the thirteenth powers, only O(T^(5/6)) parameters of size at most T produce a difference of thirteenth powers.

proposition_1_8: Every finite set of integers outside the thirteenth powers admits a thirteenth-power shift avoiding all differences of thirteenth powers.

theorem_1_1: Every integer has a unique representation as a member of one fixed set plus an integer thirteenth power.

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Pipeline-math, Erdős problem 477, six-page manuscript, version of 29 June 2026. The PDF has no individual author byline. The project README attributes proof discovery to GPT-5.5 Pro and polishing and checking to its contributors. Diyi Liu's account identifies their contribution to Problem 477. These descriptions do not establish a paper-specific author list or independent review of the manuscript.

Source identity

The copy read for this card is the unchanged upstream manuscript at commit 99d916ff32a90e77c98eb004537ccda409262346, dated 29 June 2026 00:42:25 UTC. This is the latest file-changing revision found when accessed; the initial addition was on 28 June 2026. Printed and PDF page numbers both run from 1 to 6. No journal publication or arXiv identifier was found for this manuscript in the status search. The manuscript prints no notice, and the hosting repository (https://github.com/Pengbinghui/pipeline-math, read 2026-10-02) has no license file, no license line in its README and no license in its sidebar; the term is unstated.

Result and public acceptance

Theorem 1.1 gives a set $A\subseteq\mathbb Z$ such that the translates of $\{m^{13}:m\in\mathbb Z\}$ by elements of $A$ partition $\mathbb Z$. It answers the literal all-integer formulation of Problem 477 affirmatively.

Bears on. #477: Theorem 1.1, p. 1, gives a set $A\subseteq\mathbb Z$ such that every integer is $a+m^{13}$ for exactly one pair $(a,m)\in A\times\mathbb Z$. Since $m\mapsto m^{13}$ is injective on $\mathbb Z$, taking $f(X)=X^{13}$, of degree 13, makes $f$ and $A$ an example of what the problem asks for. The other result pages (Lemma 1.4, Corollary 1.5, Proposition 1.6, Lemma 1.7, Proposition 1.8) bear on the problem only as steps of that proof. The manuscript makes no claim for other exponents or for positive or nonnegative inputs.

Thomas Bloom's signed exposition, last edited 5 September 2026 and read on 9 September, credits Price's GPT construction and independently obtained pipeline-math work. Together with the page's affirmative mathematical account, it supplies named public acceptance of the existence conclusion. Bloom expounds another exponent range, rather than giving a line-by-line review of this exact PDF. Bloom's exposition writes positive inputs, while Price's claim 154 uses nonnegative inputs. Their stronger range and this distinction are recorded on the problem page; neither changes Theorem 1.1's all-integer thirteenth-power statement. This existing public-status account is separate from the current reconstruction's verification state.

Reconstructed proof

The complete author-recorded reconstruction is distributed among Theorem 1.1 and the source-owned results Lemma 1.4, Corollary 1.5, Proposition 1.6, Lemma 1.7, and Proposition 1.8.

Lemma 1.4 excludes nonconstant rational curves on the diagonal surface when its constant is not a rational thirteenth power. Its proof includes coordinate normalization, the projective height computation, every zero-coordinate and vanishing-subsum case, and the exceptional-case line classification. Corollary 1.5 specializes the exclusion to each fixed integer outside the thirteenth powers. Proposition 1.6 combines it with an elementary coordinate bound and the external point count to obtain $O_c(T^{5/6})$ bad shifts. Proposition 1.8 takes a finite union, and Lemma 1.7 constructs pairwise disjoint translates covering every integer. Injectivity of the odd power map supplies the final uniqueness of the integer input.

The manuscript's Theorems 1.2 and 1.3 are external theorem restatements. They are represented by the canonical external interfaces below, rather than duplicate local proofs of those literature results.

External artifacts and interfaces

The three- and four-term unit bounds are the unnumbered statements recalled by Corvaja and Zannier (2011), printed p. 438, PDF p. 3, around equation (1.1), at the recalled abc and abcd interface. The journal edition is identified on the source card. Lemma 1.4 checks the normalized nonconstancy, complete-projective-line support, cardinality, height, and subsum hypotheses. The bounds are attributed to Mason-Stothers and Brownawell-Masser; their original proofs were not read or reconstructed.

The point count is Heath-Brown's Theorem 2, printed p. 1580, PDF p. 2, in the 2009 journal version of record. That version's definition counts dyadic shells. The separate arXiv:0806.4330v1 defines a whole-box count on p. 1, with Theorem 2 on p. 2. The two versions are not treated as identical; both are identified on the source card.

Source discrepancy. The manuscript's Theorem 1.3, p. 2, restates the cited journal theorem as a whole-box estimate $O_F(X^{10/k})$, with a constant depending only on $F$. The journal's Theorem 2 is a shell statement and does not provide that uniform whole-box statement. This reconstruction uses only the fixed-$c$ box bound proved locally in Proposition 1.6, with constants allowed to depend on $c$. Its shell summation and its bound for the leftover box below the shells are local reconstruction steps absent from the manuscript's proof of Proposition 1.6. This records the source discrepancy and the actual deduction used, not an author-issued erratum.

Both Heath-Brown introductions leave nonconstancy implicit in their polynomial-family definition. The reconstruction uses the positive-degree convention inferred from the source's parametrized-curve discussion and $O(R^{1/d})$ family count on journal p. 1589 (PDF p. 11), also present in arXiv v1 p. 11. The external result page and Proposition 1.6 explain this context. This is an explicitly recorded interpretation of the source context, not an author-issued correction. Corollary 1.5 excludes all such rational families, independent of their parameter-value domain.

Reading coverage and current verification

The reconstruction author read all six manuscript pages in extracted text and rendered images. This author also visually inspected Corvaja-Zannier's cover and printed pp. 437, 438, and 454, and checked the mathematical interface on p. 438 in text. For Heath-Brown, this author visually read journal pp. 1579-1580 and 1589 (PDF pp. 1-2 and 11) and arXiv v1 pp. 1-2 and 10-11; journal pp. 1580 and 1589 were also checked in extracted text. Journal p. 1589 and v1 pp. 10-11 were read only for the family convention.

Separately, the Heath-Brown source entry records another visual reading of journal pp. 1579-1581 and v1 pp. 1-2 and 10-11; the journal p. 1581 reading is not part of this reconstruction's coverage. These are distinct reading records. The statement checks and publication-page reading do not reconstruct any external proof. The remaining Corvaja-Zannier arguments, the original classical unit proofs, and Heath-Brown's full determinant-method proof receive no local proof coverage here.

The six same-source result proofs are complete. The [[diophantine_problems/pipeline_math_2026_tiling_complement/evidence/verify/compilation_review|independent compilation review]] found no material defect in their exact frozen statements, essential deductions and consumed interfaces. The coordinate normalization expands a compressed source step; the journal-shell summation and leftover-box bound are local additions absent from the manuscript proof. Both were included in the review. Attack selection was partly pre-directed; the derivations were independently performed. The six-result review is relative to the Corvaja-Zannier-recalled unit bounds and Heath-Brown's journal Theorem 2, with the recorded nonconstant-family qualification. The external proofs were not independently reviewed; no formal verification is claimed. This records independently reviewed compilation proof coverage only in that premise-relative scope, not full-manuscript acceptance, a native L-tier or a change to catalog status. Primary-source reading and public acceptance of the catalog conclusion remain separate facts.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.