Suppose $p$ and $p^2+2$ are prime numbers, prove that $p^3+2$ is also a prime number. Actually I do not know what is the relationship among square numbers, cube numbers and prime numbers
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3This is a trick problem. There's only one such possible prime $p$ so that $p^2+2$ is prime... – Thomas Andrews Jul 20 '14 at 17:34
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@ThomasAndrews what do you mean ? $3$ and $7$ seem to be such primes – Gabriel Romon Jul 20 '14 at 17:36
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Is $7^2+2$ prime? @G.T.R – Thomas Andrews Jul 20 '14 at 17:37
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Oops, my bad @ThomasAndrews – Gabriel Romon Jul 20 '14 at 17:37
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http://math.stackexchange.com/questions/269790/why-does-p28-prime-imply-p34-prime and http://math.stackexchange.com/questions/234077/prime-p-with-p28-prime – lab bhattacharjee Jul 20 '14 at 17:53
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Hint: if $p \equiv \pm 1 \mod 3$, then what is $p^2 + 2 \mod 3$? How many primes $p$ are there such that $p^2 + 2$ is also prime?
Mathmo123
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Here's a slightly different take on Matthew's approach. Multiply $p$ by $p^2+2$. You get $p^3+2p$. You know that $p^3\equiv p\bmod 3$ from Fermat's Little Theorem, therefore $p^3+2p\equiv 3p$ is a multiple of $3$. If both of your factors $p$ and $p^2+2$ are prime factors of this multiple of $3$, then one factor must be exactly $3$. Alas, $p^2+2=3$ implies $p=1$ which does not qualify as a prime, so there is only one more possibility to check.
Oscar Lanzi
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