{"schema":"vela.problem-packet.v0.1","problem":1074,"statement":"Let $S$ be the set of all $m\\geq 1$ such that there exists a prime $p\\not\\equiv 1\\pmod{m}$ such that $m!+1\\equiv 0\\pmod{p}$. Does\\[\\lim \\frac{\\lvert S\\cap [1,x]\\rvert}{x}\\]exist? What is it?Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\\not\\equiv 1\\pmod{m}$ such that $m!+1\\equiv 0\\pmod{p}$, then does\\[\\lim \\frac{\\lvert P\\cap [1,x]\\rvert}{\\pi(x)}\\]exist? What is it?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A063980","name":"Pillai primes: primes p such that there exists an integer m such that m! + 1 == 0 (mod p) and p != 1 (mod m).","terms":"23,29,59,61,67,71,79,83,109,137,139,149,193,227,233,239,251,257,269,271,277,293,307,311,317,359,379,383,389,397,401,419,","url":"https://oeis.org/A063980"},{"id":"A064164","name":"EHS numbers: k such that there is a prime p satisfying k! + 1 == 0 (mod p) and p !== 1 (mod k).","terms":"8,9,13,14,15,16,17,18,19,20,21,22,23,24,26,29,30,31,32,33,34,35,36,38,39,40,42,43,44,45,47,48,49,50,51,52,53,54,55,56,57","url":"https://oeis.org/A064164"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}