If p and p^2 +2 are primes, then p^3 +2 is a prime.

For this one I assumed that p^3 + 2 is not prime. We can easily note that p(p^2 +2) is not prime. But that's where I've been stuck.

Thanks!

2. ## Clarity?

$3| p^2+2$ for all $p>3$

3. Originally Posted by curiousmuch
If p and p^2 +2 are primes, then p^3 +2 is a prime.

For this one I assumed that p^3 + 2 is not prime. We can easily note that p(p^2 +2) is not prime. But that's where I've been stuck.

Thanks!
Hi curiousmuch.

It can be shown that $p=3$ is the only prime for which $p^2+2$ is prime. $2^2+2=6$ is not prime whereas $3^2+2=11$ is. Any prime $p\ge5$ is of the form $6n\pm1$ for some integer $n.$ But $(6n\pm1)^2+2=36n^2\pm12n+3$ is a multiple of 3 and so cannot be a prime $\ge5.$

Hence there is only one value of $p$ you need to check to see if the statement is true, namely $p=3.$ $3^3+2=29$ is indeed prime, and so the statement is true.

4. Originally Posted by Media_Man
$3| p^2+2$ for all $p>3$
Oh thanks, I was confused for a bit. Where did you get that sweet quote by the way?

5. Sorry. All prime numbers greater than 3 are of the form $6n+1$ or $6n+5$, therefore $p^2+2$ is of the form $36n+12n+1+2=3(12n+4n+1)$ or $36n+12n+25+2=3(12n+4n+9)$ . TheAbstractionist said it better. There are no primes of the form $p^2+2$, for $p$ prime, except for $p=3$, $p^2+2=11$.

*Is this theorem worded correctly?

EDIT:
Oh thanks, I was confused for a bit. Where did you get that sweet quote by the way?
Haha. "Music of the Primes" by Marcus du Sautoy. Another good one about Erdos is "The Man Who Loved Only Numbers." I've learned more about math from these kinds of books than I ever did getting a math degree in college.