Infinitely many primes $p$ with $\lfloor p\alpha\rfloor$ also prime, for irrational $\alpha>1$? (Erdős #972)
Statement
Let $\alpha>1$ be irrational. Are there infinitely many primes $p$ for which $\lfloor p\alpha\rfloor$ is also prime? Here $\lfloor\cdot\rfloor$ is the floor function, so one asks for infinitely many primes $p$ whose Beatty image $\lfloor p\alpha\rfloor$ is again prime.
Acceptance. FULLY RESOLVES: a proof (machine-checkable preferred, else a complete written proof) that for every irrational $\alpha>1$ there are infinitely many primes $p$ with $\lfloor p\alpha\rfloor$ prime, OR a proof that this fails for some irrational $\alpha>1$. ADVANCES (proof required): (a) a proof for a nontrivial explicit class of $\alpha$ (e.g. a full-measure set, all badly approximable $\alpha$, or a specific quadratic irrational), with the class clearly stated; (b) an unconditional asymptotic or lower bound for the counting function of such primes $p\le x$ for some $\alpha$; or (c) a conditional result under a stated, clearly-flagged hypothesis. Deliver the proof or formalization. Numerical prime-pair searches may motivate but do not by themselves resolve this proof-shaped question.
Background
Posed by Erdős [Er65b]; listed as open on erdosproblems.com/972 (fetched 2026-07-21, status 'open'). Context: Vinogradov [Vi48] proved that the fractional parts $\{p\alpha\}$ (over primes $p$) are uniformly distributed for every irrational $\alpha$; consequently, for every irrational $\alpha>1$ there are infinitely many primes of the shape $p=\lfloor n\alpha\rfloor$ — a prime lies in the Beatty sequence of $\alpha$ infinitely often, since $p=\lfloor n\alpha\rfloor$ iff $\{p\alpha^{-1}\}>1-\alpha^{-1}$, which holds infinitely often by uniform distribution of $\{p\alpha^{-1}\}$. The open problem demands the stronger simultaneous condition that BOTH $p$ and $\lfloor p\alpha\rfloor$ be prime — a Piatetski-Shapiro / Beatty-prime correlation out of reach of current sieve technology in general. A formalisation exists in the DeepMind formal-conjectures library. Attacker's tool: sieve for prime pairs $(p,\lfloor p\alpha\rfloor)$ for concrete $\alpha$ (e.g. the golden ratio) to gather heuristic density data, alongside circle-method / sieve arguments toward special classes of $\alpha$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #972 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #972 (formal-conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.