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While finding primes of the form $p^p+(p-1)!$ on PARI/GP, I noticed that $p$ is always prime if $p^p+(p-1)! \gt 2$ is prime. The search range was $p \le 10^5$.

Here are the solutions for $p\in\Bbb{+Z}$ for which $p^p+(p-1)!$ is prime that I got on PARI/GP:

1
2
3
11
43

Questions:

$(1)$ Will $p$ always be prime if $p^p+(p-1)! \gt 2$ is prime?

$(2)$ Are there finite primes of the form $p^p+(p-1)!$, where $p\in\Bbb{+Z}$ ? How would you prove/disprove this?

Edit: Just realized that the answer for the 1st question was obvious. But I think the second question will be much harder to answer.

Mathphile
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  • It is usually not feasible to decide whether such an expression is prime infinite many often. As in the question about $p^p+2$, I am surprised that you arrived at $p=10^5$ in PARI/GP because $p^p$ is huge for primes near $10^5$ – Peter May 09 '19 at 06:29
  • https://math.stackexchange.com/questions/3218966/is-29-the-only-prime-of-the-form-pp2/3219392#3219392 – Peter May 09 '19 at 06:40
  • Upto $p=7\ 000$ , no further prime – Peter May 09 '19 at 08:27

2 Answers2

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If $p$ is composite, then it is divisible by some prime $q<p-1$. That $q$ obviously divides both $p^p$ and $(p-1)!$

lulu
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    @quasi You put your version up first so perhaps I should be the one to delete. As an alternative, I'll just post it as "community". – lulu May 09 '19 at 00:12
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$$7901^{7901}+7900!$$ is probable prime

http://factordb.com/index.php?id=1100000001296185249

I found it with pfgw

Peter
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