Problem 946

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

Source: erdosproblems.com/946

Number theory
Divisors

Wiki page

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SolvedProved

An accepted solution exists. The statement is true.

Proved. The site credits Heath-Brown [He84] with the proof; the accepted claim is recorded on the claim page (Heath Brown, 1984), 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 (1987) and Pinner's claim page (1997).

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.
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).

Problem 946

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