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I have to find prime numbers P1, P2, P3 and P4 that satisfy the 3 equations below:

P2 = P1 + 2

P3 = P2 + 4

P4 = P3 + 8

And I'm clueless about where to start.

Which mathematical theorem/method (if any) that I could use to aid me with this question?

apple
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3 Answers3

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The prime numbers you are looking for are $p,p+2,p+6,p+14$. It can be seen that $p$ is not $3$. $p$ can not be of the form $3k+1$ because $p+2=3k+3$ which is not a prime. So $p$ must be of the form $3k+2$. For example $p=5$ or $p=17$ are solutions.

Fermat
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  • I don't understand where you got the $3k+1$ from. Let alone the point when you said $p+2=3k+3$. @Fermat – apple Feb 26 '15 at 15:22
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    You know every integer is of the form 3k, 3k+1 or 3k+2. But the only prime number of the form 3k is 3 which is not the solution (as mentioned). Now for p we have two cases 3k+1 or 3k+2. p can not be of the form 3k+1 because if p=3k+1 then p+2=3k+3=3(k+1) while the problem says p+2 is also a prime number. Hence p must be of the form 3k+2. – Fermat Feb 26 '15 at 16:02
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    I understand now. Cheers, buddy @Fermat – apple Feb 26 '15 at 16:33
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All you have to do is try a few numbers, and you'd get there quite fast.

$P_1=17$

$P_2=17+2=19$

$P_3=19+4=23$

$P_4=23+8=31$

As you can see, all those numbers are prime.

Also, as you can see, $P_4=P_1+14$, so you'd want a prime number that when added to $14$ yields a prime number, so this can be made to (probably) reduce your search when the answer is a bit bigger (not in this case).

Hasan Saad
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Start by trying some small primes as examples. I give this as an answer and not just a hint since starting with small numbers is often the best strategy for such problems.

Colin McLarty
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