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Source. Pipeline-math, Erdős problem 477, commit 99d916ff32a90e77c98eb004537ccda409262346 (29 June 2026), Theorem 1.1, p. 1 and final proof p. 6; its essential same-source arguments occupy pp. 2-6 of the manuscript. Printed and PDF page numbers agree.

Statement

Let

$$B=\{m^{13}:m\in\mathbb Z\}.$$

There exists $A\subseteq\mathbb Z$ such that, for every $n\in\mathbb Z$, there is exactly one pair $(a,m)\in A\times\mathbb Z$ satisfying

$$n=a+m^{13}.$$

Equivalently, $\mathbb Z=A\oplus B$: every integer has exactly one representation as $a+b$ with $(a,b)\in A\times B$. Zero and negative inputs are included. The conclusion is an existence result for one polynomial image, not a classification of polynomials or exponents.

Proof

Put $D=B-B$. The fully reconstructed Proposition 1.8 shows that every finite $C\subseteq\mathbb Z\setminus B$ admits a $b\in B$ with $(C-b)\cap D=\varnothing$. Its argument takes a finite union of the fixed-shift estimates

$$\#\{t\in\mathbb Z:|t|\le T,\ t^{13}-c\in D\} =O_c(T^{5/6})=o(T),\qquad c\notin B,$$

proved in Proposition 1.6. For a fixed finite $C$, that union occupies fewer than all $2T+1$ integer parameters once the integer $T$ is sufficiently large. Symmetry of $D$ changes avoidance of $t^{13}-c$ into avoidance of $c-t^{13}$.

The finite-avoidance statement is precisely the hypothesis of Lemma 1.7. That lemma enumerates $\mathbb Z$ and, at each uncovered integer $n$, applies avoidance to the finite set $n-A'$ of shifts from the previously selected translate indices. Adjoining $n-b$ covers $n$ while its difference from every old index avoids $B-B$, which proves disjointness. The lemma proves that the union of these finite sets of indices yields a set $A$ whose translates partition $\mathbb Z$. Thus every $n$ has exactly one pair $(a,b)\in A\times B$ with $n=a+b$.

For each $b\in B$ there is an integer $m$ with $b=m^{13}$. If two integers had the same thirteenth power, strict increase of the odd power function on $\mathbb R$ would make them equal. Therefore the unique pair $(a,b)$ corresponds to exactly one pair $(a,m)$, proving the theorem.

Complete proof scope and external premises

The proof is distributed over the linked same-source results rather than duplicated here. Their dependency order is Lemma 1.4, Corollary 1.5, Proposition 1.6, Proposition 1.8, and finally Lemma 1.7 and the deduction above. Lemma 1.7 is an independent combinatorial criterion.

Lemma 1.4 proves the rational-curve exclusion by handling every active coordinate count and vanishing-subsum case. It uses only the three- and four-term genus-zero bounds recalled on Corvaja-Zannier (2011), printed p. 438, at the external unit-bound interface. The normalization, projective height calculation, rationality of the resulting constants, and exceptional-line classification are included. Corollary 1.5 specializes that result to the surfaces used in counting.

Proposition 1.6 uses the journal version of Heath-Brown's Theorem 2, printed p. 1580. It proves the coordinate bound and excludes the nonconstant polynomial families. The journal-shell summation and the bound for the leftover box below the shells, where the theorem's range condition need not hold, are local reconstruction steps absent from the manuscript's Proposition 1.6 proof. They supply only the fixed-$c$ whole-box bound.

The manuscript instead invokes its Theorem 1.3, a whole-box restatement with an $F$-only constant that the cited journal shell statement does not provide. Its proof does not make our separate comparison between the journal's shell definition and arXiv v1's whole-box definition. This reconstruction uses the journal theorem and the local fixed-$c$ deduction, not the stronger manuscript restatement. The positive-degree family convention is inferred from source context, including journal p. 1589 (PDF p. 11), as explained in Proposition 1.6 and the external result page. No author-issued erratum is claimed.

These external theorem statements were checked at the identified primary pages. Their proofs are external literature premises, not included local proof coverage. No original Brownawell-Masser or Mason-Stothers proof is claimed to have been read.

Current verification

This complete source-proof reconstruction, including its essential same-source results, received [[diophantine_problems/pipeline_math_2026_tiling_complement/evidence/verify/compilation_review|independent compilation review]]. No material defect was found in the exact frozen statement, every essential deduction or the consumed interfaces. All six manuscript pages were read in extracted text and rendered images. 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.

The source digest pins the manuscript and external versions and states the actual reading depth. Public acceptance of the catalog conclusion is a separate record; it is not a review of these pages.

Bears on. Taking $f(X)=X^{13}$, of degree 13, proves the existence proposition for the full image $f(\mathbb Z)$ in Problem 477. Uniqueness of a polynomial value and uniqueness of its input coincide for this injective polynomial. No claim about the other exponent ranges or positive-input variants recorded on that problem page is made here.