2012-09-25

Bengt, Fermat, and Mersenne

Fermat had a lot of clever ideas in mathematics, but the Fermat primes were maybe not his brightest one. He claimed that any number on the form 2^(2^n)+1 is a prime, and made it plausible by showing it to be true for all n up to 4. Sadly, like so many other beautiful ideas in science, it was ruined by the Germans; Euler showed that it is not true for n = 5. (Actually, Euler was from Switzerland, but never mind that now.) A bunch of other Fermat numbers have been tested, but so far none of them have turned out to be primes. It has been proven that at least up to n = 32, there are no more Fermat primes. What a bust. Another guy who thought about primes was Mersenne, and his idea was 2^n-1, where n is a prime. Those things are not always primes either, but at least it happens fairly often. Bengt has what he thinks sounds like an awesome idea. You might call them the Bengt numbers, but to be honest he's not quite sure enough that they're going to work out that he wants to name them after himself. After all, you don't want to end up looking silly like Fermat. Here's his idea: n! is divisible by all the numbers up to n. Therefore n!+1 can't be divisible by any of them, so it must be a prime. The same is true for n!-1, so you get two primes for the price of one. Bengt is happy having come up with such a brilliant idea, so he goes off to treat himself to some chocolate, and forgets all about the whole thing.

  1. First, we'll play around with the Mersenne numbers for a bit. Find the first that isn't a prime.
  2. Now let's see if we can help poor Fermat out. So 2^n+1 didn't work, and 2^(2^n)+1 doesn't work. What about 2^(2^(2^n))+1? Actually, never mind that - let's go a little crazy and say 2^(2^(2^(2^(2^(2^n)))))+1. Are those all primes?
  3. It's probably a good idea that Bengt didn't name those numbers after himself. Explain where he got it wrong, and find a counterexample.
  4. What if n is always prime? Counterexample?
  5. Even if it doesn't work for every n, it might still work for infinitely many n. Does it? Are there infinitely many n for which n!+1 and n!-1 are both primes?
  6. (hard) Is there any function that does what Bengt want - that is, there are infinitely many n such that f(n)+1 and f(n)-1 are primes?