Problem Statement
Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0<a<n$ and\[\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?\]
Categories:
Number Theory Primes
Progress
Erdős and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\liminf$ by $\limsup$.Source: erdosproblems.com/428 | Last verified: January 15, 2026