Find all primes p such that 2^p + p^2 is also prime.
Prove there are no
others.
This has already been solved by others, but here is my solution.
First, note that p=2 is not a solution.
For any odd number (prime or not), 2^odd is 2 mod 3.
Odd numbers themselves can be 0, 1, or 2 mod 3.
But:
squaring a 2 mod 3 number produces a 1 mod 3 number.
squaring a 1 mod 3 number produces a 1 mod 3 number.
squaring a 0 mod 3 number produces a 0 mod 3 number.
So for all odd numbers that are not divisible by 3, the formula (in mod 3) produces 2+1=3 which is 0 mod 3 and not a prime.
That is, not a prime except for 3 itself which is the only prime which is divisible by 3.
So 3 is the only answer.
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Posted by Larry
on 2025-05-29 10:34:24 |
solution
