Determine whether every irrational real number alpha has infinitely many primes ell such that |alpha ell|<ell^-1, where |x| denotes the distance from x to the nearest integer.

Erdős–Mahler question on Diophantine approximation along primes

The Erdős–Mahler question concerns Diophantine approximation restricted to prime indices. The retrieved material does not identify its proposer or original date.

September 2026 prime-progress preprint

The preprint Diophantine approximation with primes in an arithmetic progression claims the bound $\|\alpha\ell-\beta\|\ll_v\ell^{-1/4}\log^8\ell$ in arithmetic progressions and constructs an uncountable family of parameters having infinitely many approximations of order $\ell^{-1}$ . This is a substantive advance, but it does not classify all parameters or settle the full question; the claims remain unverified.

Current status (as of September 2026): The full question remains open, with a substantial but unverified claimed advance from the new prime-arithmetic-progression preprint.

Erdős–Mahler question on Diophantine approximation along primes

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