Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 946

claims/: The 3 claim pages of Problem 946, one per claimant's result; the problem's standing derives from them.

Statement. Are there infinitely many $n$ such that $\tau(n)=\tau(n+1)$ , where $\tau$ is the divisor function?

Status. Proved. The site credits Heath-Brown [He84] with the proof; the accepted claim is recorded on the claim page, and Hildebrand's 1987 count and Pinner's 1997 theorem for every shift, each of which answers the question again, on Hildebrand's claim page and Pinner's claim page.

Source. erdosproblems.com/946, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #946, https://www.erdosproblems.com/946.

References.

[EPS87] Erdős, Paul and Pomerance, Carl and Sárközy, András, On locally repeated values of certain arithmetic functions. III. Proc. Amer. Math. Soc. (1987), 1-7.
[ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function $d(n)$ . Proc. London Math. Soc. (3) (1952), 257-271.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B18 "Solutions of $d(n)=d(n+1)$ ", printed pp. 111--112, where the book reports the Spiro, Heath-Brown and Pinner results and the counts of Erdős, Pomerance and Sárközy and of Hildebrand. Library home: guy_2004_unsolved_problems_number_theory.
[He84] Heath-Brown, D. R., The divisor function at consecutive integers. Mathematika (1984), 141-149.
[Hi87] Hildebrand, Adolf, The divisor function at consecutive integers. Pacific J. Math. (1987), 307-319.
[Pi97] Pinner, Christopher G., Repeated values of the divisor function. Quart. J. Math. Oxford Ser. (2) (1997), 499-502.
[Sp81] Spiro, Claudia Alison, THE FREQUENCY WITH WHICH AN INTEGRAL-VALUED, PRIME-INDEPENDENT, MULTIPLICATIVE OR ADDITIVE FUNCTION OF N DIVIDES A POLYNOMIAL FUNCTION OF N. (1981).
[TaTe25] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers. arXiv:2512.01739 (2025).

Formalization. Statement in formal-conjectures, which, at the commit of 18 September 2026 linked here, carries a formal_proof attribute pointing at Erdos946.lean in Boris Alexeev's lean-proofs repository, a Lean proof that follows Heath-Brown's method and is linked, pinned, from his claim page; this corpus has not built it.

Current assessment

The question is whether $\tau(n)=\tau(n+1)$ holds for infinitely many $n$ . Heath-Brown's 1984 theorem answers yes, with at least $cx/(\log x)^{7}$ such $n\le x$ ; the problem's standing derives from his claim page, which is accepted on the curator's credit and the journal publication. Hildebrand's lower bound $\gg x/(\log\log x)^{3}$ (claim page; card) and Pinner's theorem [Pi97] that $\tau(n)=\tau(n+k)$ holds infinitely often for every shift $k\ge1$ (claim page) each answer the question again. The other later results leave the answer unchanged: the upper bound $\ll x/\sqrt{\log\log x}$ of Erdős, Pomerance and Sárközy (card), and the theorem of Tao and Teräväinen [TaTe25] (card), whose Theorem 1.7 (arXiv v2) shows that, for $x$ in a set of logarithmic density one, the proportion of $n\le x$ with $\tau(n)=\tau(n+1)$ is $(c_\tau+O((\log\log x)^{-c}))/(2\sqrt{\pi\log\log x})$ , where $c>0$ is an absolute constant and $c_\tau$ (the paper's Definition 1.5) is the limiting probability that $\tau(n+1)/\tau(n)$ is a power of two, empirically about $0.4888$ (its Remark 1.6): the conjectured asymptotic for almost all scales; the card's local check covered its Remark 1.4 and not that theorem. The question goes back to Erdős and Mirsky [ErMi52], and Guy's collection [Gu04] reports the same chain of results. No proof has been reproduced or reviewed here, and no literature search beyond the site's page is recorded.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (14)

Heath-Brown's infinitely many equal divisor counts at n and n plus 1 Hildebrand's count of equal divisor counts at consecutive integers Pinner's equal divisor counts at every fixed shift problems/divisors/E0946/claims Divisors and Multiples arithmetic_functions/erdos_1987_locally_repeated_values_certain_arithmetic_functions Theorem 2.1 (p. 2): consecutive integers with the same number of distinct prime factors number O(x/√(log log x))arithmetic_functions/tao_2025_quantitative_correlations_problems_prime_factors_consecutive Divisors and Multiples divisors/erdos_1952_distribution_values_divisor_function The divisor function at consecutive integers Lemma 1 (p. 309): Heath-Brown's Key Lemma on k-tuples a_1 < ... < a_k with a_j - a_i dividing (a_i, a_j)Theorem 1 (p. 307): d(n) = d(n+1) for at least c x (log log x)^{-3} integers n <= x number_theory/guy_2004_unsolved_problems_number_theory

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