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!
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 $\displaystyle p=3$ is the only prime for which $\displaystyle p^2+2$ is prime. $\displaystyle 2^2+2=6$ is not prime whereas $\displaystyle 3^2+2=11$ is. Any prime $\displaystyle p\ge5$ is of the form $\displaystyle 6n\pm1$ for some integer $\displaystyle n.$ But $\displaystyle (6n\pm1)^2+2=36n^2\pm12n+3$ is a multiple of 3 and so cannot be a prime $\displaystyle \ge5.$
Hence there is only one value of $\displaystyle p$ you need to check to see if the statement is true, namely $\displaystyle p=3.$ $\displaystyle 3^3+2=29$ is indeed prime, and so the statement is true.
Sorry. All prime numbers greater than 3 are of the form $\displaystyle 6n+1$ or $\displaystyle 6n+5$, therefore $\displaystyle p^2+2$ is of the form $\displaystyle 36n+12n+1+2=3(12n+4n+1)$ or $\displaystyle 36n+12n+25+2=3(12n+4n+9)$ . TheAbstractionist said it better. There are no primes of the form $\displaystyle p^2+2$, for $\displaystyle p$ prime, except for $\displaystyle p=3$, $\displaystyle p^2+2=11$.
*Is this theorem worded correctly?
EDIT: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.Oh thanks, I was confused for a bit. Where did you get that sweet quote by the way?