Problem 828

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Statement. Is it true that, for any $a\in\mathbb{Z}$, there are infinitely many $n$ such that

$$\phi(n) \mid n+a?$$

Status. Open.

Source. erdosproblems.com/828, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #828, https://www.erdosproblems.com/828.

References.

• [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 B37 "Does $\phi(n)$ properly divide $n-1$?" is on pp. 142--143; Graham's conjecture that for all $k$ there are infinitely many $n$ with $\phi(n)\mid(n-k)$ is on p. 143. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

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• guy_2004_unsolved_problems_number_theory