suppose p and p square + 8 are prime. show that P cube +4 is also prime

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- Oct 20th 2011, 02:07 AMclaudio411primes
suppose p and p square + 8 are prime. show that P cube +4 is also prime

- Oct 20th 2011, 08:27 AMalexmahoneRe: primes
Rewriting the question using LaTeX: Suppose $\displaystyle p$ and $\displaystyle p^2+8$ are prime, show that $\displaystyle p^3+4$ is also prime.

All primes $\displaystyle p$ other than 3 are of the form $\displaystyle 3k\pm 1$.

$\displaystyle p^2+8=(3k\pm 1)^2+8$

$\displaystyle =9k^2\pm 6k+9$, which is divisible by 3.

So, 3 is the only prime $\displaystyle p$ such that $\displaystyle p^2+8$ is also prime. Then, $\displaystyle p^3+4=31$ is also prime.