p factorial minus p primorial

Question

Let be k# the primorial function. Let be $p$ a prime.

Is $19$ the only prime $p$ such that p!-p# is divisible by $p^2$? And why?

Answer 1

Because of Wilson's theorem the given condition is equivalent to $\ p$# $\equiv -p\mod p^2\ $

Upto $10^6$ , the following primes do the job ($\ p=2\ $ and $\ p=3\ $lead to $\ 0\ $, so might better be ruled out) :

? s=1;forprime(p=1,10^6,s=s*p;if(Mod(s,p^2)==-p,print1(p," ")))
2 3 19 1471 3001
?

Answer 2

Probably not. This is the one of many questions you have asked about particular numerical coincidences you observed about primes. Most of those have been closed. There are infinitely many primes and infinitely many possible coincidences so if you keep looking you will stumble across many.