Let $p$ be a prime number,and $p^2+2$ is also prime, prove that $p=3$.

Question

I tried to solve this question but without a success: Let $p$ be a prime number,and $p^2+2$ is also prime, prove that $p=3$.

I tried to show $p^2+2$ as a product of numbers and then to show that $p=3$ is the only option that allows it to be prime. but I didn't find that presentation.

I would like to get help with this question, thanks

Hint: All primes except for $2$ and $3$ are of the form $6n\pm1$. — Lucian, Apr 18 '15 at 12:50

score 5 · Answer 1 · edited Apr 18 '15 at 12:54

5

if $p$ is not $3$ then $\gcd(p,3)=1$ so $3$ divides $(p-1)(p+1)$ hence $3$ divides $p^2-1+3=p^2+2$ so $p^2+2$ is not a prime

edited Apr 18 '15 at 12:54

mathlove

139,939

answered Apr 18 '15 at 12:34

Elaqqad

13,725

Let $p$ be a prime number,and $p^2+2$ is also prime, prove that $p=3$.

1 Answers1