Problem 380

We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\prod_{u\leq m\leq v}m$ occurs with an exponent greater than $1$ . Let $B(x)$ count the number of $n\leq x$ which are contained in at least one bad interval. Is it true that

B(x)\sim \#\{ n\leq x: P(n)^2\mid n\},

where $P(n)$ is the largest prime factor of $n$ ?

Source: erdosproblems.com/380

Number theory

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An accepted solution exists. The statement is true.

Proved, the site's label (PROVED, page last edited 10 April 2026): Tao's 2026 preprint proves the asymptotic with a relative error of a power of the logarithm, and the site's curator records it as the proof. The accepted claim is Tao 2026; a Lean development that proves the asymptotic by its own route is the pending claim Alexeev 2026.

Problem 380

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