Find all primes p such that 2^p + p^2 is also prime.
Prove there are no
others.
First, we see that it is not true for p=2, and it is for p=3. The only possible answer is 3, since for all primes p>3, 2^p+p^2 is a multiple of 3.
To see this consider (for p>3):
(p^2 -1) + (2^p + 1) =
(p-1)(p+1) + (2^p + 1)
Now since p can't be a multiple of 3, either (p-1) or (p+1) must be a multiple of three, so (p-1)(p+1) is a multiple of 3. Also, since p is odd, 2^p + 1 is a multiple of 3, so the whole number must be a multple of 3.
Edited on November 19, 2004, 9:57 pm
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Posted by SteveH
on 2004-11-19 21:55:13 |
Solution with mini-proof
