Problem #972: Let $\alpha>1$ be irrational

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Progress

Vinogradov [Vi48] proved that the sequence $\{p\alpha\}$ is uniformly distributed for every irrational $\alpha$, and hence there are infinitely many primes $p$ of the shape $p=\lfloor n\alpha\rfloor$ for every irrational $\alpha>1$. Indeed, this occurs if and only if\[\frac{p}{\alpha}\leq n<\frac{p+1}{\alpha},\]which is true if and only if $\{p\alpha^{-1}\}>1-\alpha^{-1}$, which happens infinitely often by the uniform distribution of $\{p\alpha^{-1}\}$.