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Claim. The answer to Problem 946 is yes. D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), no. 1, 141--149, proves that $\tau(n)=\tau(n+1)$ for infinitely many $n$, and more precisely that for some constant $c>0$ and all large $x$ at least

$$c\,\frac{x}{(\log x)^{7}}$$

integers $n\le x$ satisfy it. The paper sharpens the method by which Spiro had shown that $\tau(n)=\tau(n+5040)$ holds infinitely often, bringing the shift down to $1$. The journal issue is dated June 1984 and no day is recorded, so this page carries the first of that month.

Context. Hildebrand raised the lower bound to $\gg x/(\log\log x)^{3}$ in 1987 (claim page), and Pinner's 1997 paper carries Heath-Brown's method over to every shift $k\ge1$ (claim page); each answers the question again. Erdős, Pomerance and Sárközy proved the upper bound $\ll x/\sqrt{\log\log x}$ in 1987 (card).

Depends on. No page of this wiki.

Formalization. The file Erdos946.lean in Boris Alexeev's lean-proofs repository, linked above at a pinned commit and added to the repository on 26 August 2026, proves erdos_946, that the set of $n$ with $\tau(n)=\tau(n+1)$ is infinite, without sorry. Its header says the proof follows Heath-Brown's key-and-sieve method with an explicit sixteen-element key and deliberately loose sieve parameters, cites this paper, and names no author; its docstring attributes the affirmative answer to Heath-Brown. The formal-conjectures statement file of the problem points at it through a formal_proof attribute since 18 September 2026. This corpus has not built or audited it, so no formalized evidence is listed.

Acceptance. Thomas Bloom, the site's curator, labels the problem proved and credits Heath-Brown's paper for the proof on the problem page, last edited 2 February 2026; that credit is the reviewed evidence. The paper is a refereed article in Mathematika, the refereed evidence. The paper is not held in the library and its proof has not been reproduced here; the count is recorded as the problem page and the Erdős, Pomerance and Sárközy card state it.