Prove that $n$ and $n^2+2$ are prime if and only if $n = 3 $

Question

Here is where I started

Note that $n$ must be a prime number Note that $n = 2$ is not a solution Note that $n = 3$ is a solution Assume there is some prime number $p$ such that $p > 3$ and $p$ and $p^2 + 2$ are prime

$p$ can be written as $3+k$ where $k$ is an even natural number

hence,$$ p^2+2 = (3+k)^2+2 = 9+6k+k^2+2$$ $$= k^2 + 6k + 11$$ I am stuck, I don't know where to go even though I know k is even...

Hint: any integer can be written in the form $3k$ or $3k+1$ or $3k+2$. Check what happens in each case. — Yathi, Dec 28 '23 at 13:39
Does this answer your question? For which primes p is $p^2 + 2$ also prime? - found through using an Approach0 search. Note there are also other duplicates, e.g., Suppose that both $p$ and $p^2 +2$ are prime numbers. Show that $p=3$., and also very similar ones where it asks to show $p^3+2$ is also prime, e.g., Squares, Cubes and Prime numbers. — John Omielan, Dec 28 '23 at 17:09

score 6 · Answer 1 · answered Dec 28 '23 at 13:45

6

Now I see, $p =3k+1, 3k-1$

$p≡ \pm1 \mod 3$

$p^2≡ 1 \mod 3$

$p^2+2 ≡ 0 \mod 3$

$p^2+2$ is always divisible by 3, hence it cannot be prime

answered Dec 28 '23 at 13:45

uggupuggu

396

1

Please strive not to post more (dupe) answers to dupes of FAQs. This is enforced site policy, see here. It's best for site health to delete this answer (which also minimizes community time wasted on dupe processing.) – Bill Dubuque Dec 28 '23 at 17:47
What does Dupe mean – uggupuggu Dec 29 '23 at 07:13
Abbreviation of duplicate. – Bill Dubuque Dec 29 '23 at 07:15
Should I delete the question then – uggupuggu Dec 29 '23 at 07:17
That would be best, since we already have way too many questions like this (thee more dupes the harder it is for users to locate the "best" answers). – Bill Dubuque Dec 29 '23 at 07:23

Prove that $n$ and $n^2+2$ are prime if and only if $n = 3 $

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