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        riddles >> putnam exam (pure math) >> f(n) | 2^n - 1
        
(Message started by: Eigenray on Mar 15^th, 2008, 2:36pm)

Title: f(n) | 2^n - 1
Post by Eigenray on Mar 15^th, 2008, 2:36pm

Determine all integer polynomials f such that f(n) | 2ⁿ - 1 for all n > 0.

Title: Re: f(n) | 2^n - 1
Post by Obob on Mar 15^th, 2008, 4:49pm

If there are infinitely many Mersenne primes, then [hide] either f(n) = 1 or f(n) = -1 for all n [/hide] . Since we don't know if there are infinitely many Mersenne primes or not, if the problem has an answer then this is it.

Title: Re: f(n) | 2^n - 1
Post by Eigenray on Mar 20^th, 2008, 5:07pm

Yes, that is the answer. Hint: What do you know about f(p) when p is prime?

Title: Re: f(n) | 2^n - 1
Post by Eigenray on Jul 16^th, 2008, 3:01pm

Looks like I forgot about this one. I guess that was a bad hint. I should have said: what do you know about the prime factors of f(p) when p is prime?

Title: Re: f(n) | 2^n - 1
Post by Aryabhatta on Mar 14^th, 2009, 1:08pm

Almost a year!

Title: Re: f(n) | 2^n - 1
Post by Eigenray on Mar 14^th, 2009, 2:45pm

Well, what do you know about the prime factors of f(p) when p is prime?

Further hint: [hide]Dirichlet[/hide]

Title: Re: f(n) | 2^n - 1
Post by Obob on Mar 14^th, 2009, 4:15pm

If q is a prime divisor of 2^p-1, for p prime, then 2^p = 1 (mod q). Hence p is a multiple of the order of 2 in (Z/qZ)*, and since p is prime actually p is the order of 2 in (Z/qZ)*. This implies p | q-1, so kp = q-1 for some k, and q = kp+1 for some k.

I'm not sure where this is going.

Title: Re: f(n) | 2^n - 1
Post by Eigenray on Mar 14^th, 2009, 6:51pm

So q can't divide very many values f(p), can it?

Title: Re: f(n) | 2^n - 1
Post by Obob on Mar 15^th, 2009, 4:47pm

Ah. Suppose the prime q other than p divides f(p) for some prime p, so that f(p) = 0 (mod q). In Z/qZ, we have the identity f(p+kq) = f(p); see this at the level of monomials by applying the binomial theorem. Letting k vary, the expression u = p+kq is prime infinitely often by Dirichlet's theorem since p != q, so q divides f(u) for infinitely many primes u. But q > u for all such primes u by my previous post, a contradiction.

This forces the only prime factor of f(p) to be p for each prime p. But again by my previous post p never divides 2^p-1. Thus in fact f(p) = +- 1 for each p, from which we easily conclude that either f(n) = 1 for all n or f(n)= -1 for all n.

That was a nice problem. Number theory isn't really my thing, but this was fun to think about. (I do algebraic geometry, preferably in characteristic 0)