(3^n+1)/2 is prime. A finite sequence?

[Don Reble]

> (3^n+1)/2 is prime for n=1,2,4,16,32,64. Can it be shown that no other
> (larger) sequence terms exist? The only candidates for this have the
> form n = 0 mod 32.
  ...
> No more primes of the form (3^(32*m)+1)/2 found up to m=800.

I hope you didn't check _all_ the multiples of 32. :-)

Let Q be the largest odd factor of N, so E=N/Q is a power of two.
Then 3^N+1 = (3^E)^Q+1^Q, and divisible by 3^E+1. If Q>1 then 3^E+1
is a proper factor; and since 3^E+1 is even, the (proper) cofactor
divides (3^N+1)/2 properly.

Thus if N is not a power of two, Q>1 and (3^N+1)/2 is composite.
And it's also composite for N=2^{3,7,8,9,10,11,12,13,14,15, and 16}.
In each case, 2 is a strong-test witness.

I wonder, does 2^2^K+1 always divide 3^2^(2^K-1)+1?

--
Don Reble       djr at nk.ca