A007689: all terms composite?

[Dean Hickerson]

zak seidov asked when a(n) = 2^n + 3^n is prime:

> a(n) are primes for n=0,1,2,4.
> My Q: what about other terms a(2k), 
> are they all composite?

This is similar to the case of Fermat primes.  If k is odd, then a(nk) is
divisible by a(n), since:

    a(nk) = (2^n)^k + (3^n)^k

          = (2^n + 3^n) [(2^n)^(k-1) - (2^n)^(k-2) (3^n) + - ... + (3^n)^(k-1)]

So the only possible primes in the sequence are a(0) and a(2^n) for n>=1.
I've checked that a(2^n) is composite for 3 <= n <= 15.  As with Fermat
primes, a probabilistic argument suggests that there are only finitely many
primes in the sequence, but I doubt that anyone can prove it.

Dean Hickerson
dean at math.ucdavis.edu