A064539 All...are divisible by 3 but why?

[Hugo Pfoertner]

David Harden wrote:
> 
> >From: zak seidov <zakseidov at yahoo.com>
>

[...]

> >the more(?) interesting Q is:
> >are really cases n=1 and 28
> >the only two(?) ones(??)
> >making 3^(2n)+(2n)^3 prime?
> >see A109215 (pending, and, sorry, again "base"),
> >thanks, zak
> 
> This question is definitely more interesting, and my personal guess is that there are infinitely many n such that 3^(2n) + (2n)^3 is prime. How far has this sequence been computed? It is interesting that the residue of this quantity modulo the prime p will require the knowledge of p modulo the binomial coefficient (p  2) whenever 3 is a primitive root modulo p or when p == 11 (modulo 12) and 3 has order (p-1)/2 modulo p.
> 

I checked (using PFGW) up to n=3500 and found no further prime. Seem to
to be rare beasts in this case ;-)

Hugo Pfoertner