Problem 676

Erdős Problem #676: declared status 'open'. Formalized: no. Is every sufficiently large integer of the form$$ap^2+b$$for some prime $p$ and integer $a\geq 1$ and $0\leq b<p$? Current best: Most generally, given some infinite set $A\subseteq \mathbb{N}$ and function $f:A\to \mathbb{N}$ one can ask for sufficient conditions on $A$ and $f$ that guarantee every large number (or almost all numbers) can be written as$$am^2+b$$for some $m\in A$ and $a\geq 1$ and $0\leq b<f(m)$. In another direction, one can ask what is the minimal $c_n$ such that $n$ can be written as $n=ap^2+b$ with $0\leq b<c_np$ for some $p\leq \sqrt{n}$. This problem asks whether $c_n\leq 1$ eventually, but in [Er79d] Erd\H{o}s suggests that in fact $\limsup c_n=\infty$. Prize: no. OEIS: A390181. Tags: number theory.