Erdős Problem #493

The problem

Does there exist a $k$ such that every sufficiently large integer can be written in the form

$$\prod_{i=1}^k a_i - \sum_{i=1}^k a_i$$

for some integers $a_i\geq 2$?

Statement quoted from erdosproblems.com/493, maintained by Thomas Bloom. Original source: [Er61,p.238]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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