Problem 964

Let $\tau(n)$ count the number of divisors of $n$ . Is the sequence

\frac{\tau(n+1)}{\tau(n)}

everywhere dense in $(0,\infty)$ ?

Source: erdosproblems.com/964

Number theory
Divisors

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SolvedProved

An accepted solution exists. The statement is true.

Proved. The site labels the problem PROVED (LEAN) and credits Eberhard's proof, which answers the question affirmatively; his stronger theorem says every positive rational occurs infinitely often. The community Lean file posted in the forum thread contains a complete formal proof conditional on a formalized GGPY proposition, which that file does not prove. The accepted claim is recorded on Eberhard's claim page (2025).

Problem 964

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Proof claims

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