find all primes

**mandy123** · May 6th 2008, 01:04 PM

find all primes of the form (2^2^n) + 5 where n is a nonnegative integer

**PaulRS** · May 6th 2008, 02:11 PM

It is always divisbile by 3 for n>0 (and $2^{2^n}+5\neq{3}$ for all nonnegative integers n)

Just note that: $2^{2^n}+5\equiv{(-1)^{2^n}+2}(\bmod.3)$

If $n>0$ we have that $2^n$ is even and therefore $(-1)^{2^n}=1$ thus: $2^{2^n}+5\equiv{3}\equiv{0}(\bmod.3)$

If n=0 we have: $2^{1}+5=7$ that is prime

Therefore the only prime in that sequence is 7

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