## Factorials, Primes, and Long Life

Today my age rolls over from a factorial to a square, and this will happen again on December 3, 2105. How old am I?

It occurred to me a little while ago to try to find all the times this could ever happen. A little calculation shows that $6!+1$ is not a square, but that $7!+1 = 71^2$.

After that, I couldn’t find any more factorials that were one less than a square. I set up a quick computer program to search, and didn’t find any looking up to $n = 200$. So I began to wonder whether there were any more, and if not, how could I prove it?

One of the most famous proofs in number theory states that there are infinitely many primes, because if not, then there must be a largest prime $p$. But then $p!+1$ has no divisors other than one, since all of the primes have remainder one when they divide it. This is nonsense – of course $p!+1$ has a divisor – itself! So by contradiction there are not a finite number of primes. This suggests that perhaps we’d expect $n!+1$ to be prime much of the time. Or, since we could have used $p!-1$ in the above argument, that $n!-1$ may frequently be prime. And, since $n!+1$ and $n!-1$ differ by two, maybe there are infinitely-many twin primes?

No one knows how to answer these simple questions. The numbers $n! \pm 1$ are called factorial primes, and we’ve found $42$ of them so far. That’s a long way short of infinity. On the other hand, who knows? We might already have them all.

The factorials one less than a square have also been studied, but the catalog here is much less extensive. In fact, I unwittingly discovered the entire thing in my first investigation. These are called Brown Numbers, and they are the solution to Brocard’s Problem. There are no more up to $n=10^9$, and people who professionally think about these things are strongly suspicious that there are no more at all. It has been proven that there are finitely-many. That doesn’t mean there couldn’t be a fourth sitting way, way out there, waiting for us to find it. And if we did, that doesn’t mean there couldn’t be a fifth…

It looks like if I want an answer to this question, I’m either going to have to invent some new mathematics, or wait for someone else to, or else live a very, very long time.