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

Thanks very much!!:)

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- Sep 26th 2006, 05:46 PMsuedenationPrime number of a form
Find all primes of the form 2^(2^n)+5, where n is a nonnegative integer

Thanks very much!!:) - Sep 26th 2006, 08:36 PMThePerfectHacker
Only n=0

Because if n>0 then you have

none, all these numbers are divisible by three.

Proof

~~~

We know that,

2 = -1 (mod 3 )

Then,

2^(2^n) = (-1)^(2^n) (mod 3)

But,

(-1)^(2^n)=1 because exponent is even.

Thus,

2^(2^n) = 1

Add 5 to both sides,

2^(2^n)+5 = 1+5=6=0 (mod 3)

Q.E.D.