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 is not a square, but that .
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 . 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 . But then has no divisors other than one, since all of the primes have remainder one when they divide it. This is nonsense – of course has a divisor – itself! So by contradiction there are not a finite number of primes. This suggests that perhaps we’d expect to be prime much of the time. Or, since we could have used in the above argument, that may frequently be prime. And, since and differ by two, maybe there are infinitely-many twin primes?
No one knows how to answer these simple questions. The numbers are called factorial primes, and we’ve found 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 , 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.