Thread: Prime number of a form

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

Thanks very much!!

Originally Posted by suedenation

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

Thanks very much!!

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.