3

It is still unknown if $n!+1=m^2$ has only 3 solutions, i.e. for $n=4,5,7$.

$n!+1$ being a perfect square

https://en.wikipedia.org/wiki/Brocard%27s_problem

What about my problem:

Are there finitely or infinitely many numbers $n$ such that $n!+1$ is divisible by a perfect square greater than 1?

It is also interesting if $n!+1$ is squarefree for infinitely many $n$.

For $n\le 100$, number $n!+1$ is:

  • prime for $n=1,2,3,11,27,37,41,73,77$

  • perfect square for $n=4,5,7$

  • divisible by a square (while not being a perfect square) for $n=12,23$

  • composite squarefree for other $n$'s

tong_nor
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    Other examples I found so far : $$563^2\mid 562!+1$$ $$613^2\mid 229!+1$$ – Peter Aug 25 '19 at 11:54
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    If infinite many WIlson primes (https://en.wikipedia.org/wiki/Wilson_prime) exist which is conjectured the answer to your question is "infinite many n" – Peter Aug 25 '19 at 11:56
  • Thanks! Did you factorize $n!+1$ up to $n=562$? Are all others (for $n=101,\dots 561$ except 229 composite squarefree? – tong_nor Aug 25 '19 at 12:10
  • No, with factordb you can look up the factorizations of $n!+1$ upto a quite large limit. – Peter Aug 25 '19 at 12:11
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    But for a further example, the prime factor must exceed $10^5$ – Peter Aug 25 '19 at 12:12
  • I just completed the $10^5$-range – Peter Aug 25 '19 at 12:13
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    Unfortunately, determining whether a number is squarefree is to current knowledge not easier in general than factoring. For $n\le 1\ 000$ , for another pair the prime factor must exceed $10^7$ – Peter Aug 25 '19 at 12:23
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    The first hole in factordb is at $n=140$ , so we cannot be absolutely sure in the case of $140!+1$ until someone factored it – Peter Aug 25 '19 at 12:35
  • that's a nice tool (factordb - I didn't knew about this, I was using WolframAlpha, it factorizes up to $100!+1$ and fails to factorize for the first time at $101!+1$) – tong_nor Aug 25 '19 at 12:43
  • Then, I assume Wolfram Alpha knows the factors because within the time-limit there, the factorization would probably not be completed for all $n$ upto $100$ – Peter Aug 25 '19 at 12:44
  • Do you have yafu and/or pfgw and a fast computer ? You could help me in several factorization- and prime-finding-projects. pari/gp is of (limited) value as well. – Peter Aug 25 '19 at 12:46
  • unfortunately no, I'm using a 5-years-old laptop, almost all computations I perform (usually for own curiosity, in own investigations like this) I do with WolframAlpha, sometimes I ask my friend who has a stronger machine and a student version of Mathematica, but I don't have an access to a really powerful computer. – tong_nor Aug 25 '19 at 13:01
  • https://math.stackexchange.com/questions/3513224/a-possible-solution-for-erdõs-conjecture-n2-1-k-and-brocards-problem – Charles Kusniec Jan 18 '20 at 13:42
  • What was there under this link? This question was removed... – tong_nor Mar 02 '20 at 13:31

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