For a positive integer n, if both n and n + 2 are prim

[Mellie]

For a positive integer [tex]n[/tex], if both [tex]n[/tex] and [tex]n + 2[/tex] are prime, then they are known as twin primes. For example, 59 and 61 are twin primes. Whether or not there are an infinite number of twin primes is a famous unsolved problem in number theory.

Find all positive integers [tex]n[/tex] such that [tex]n[/tex], [tex]n + 2[/tex], and [tex]n + 4[/tex] are all prime.

(Write your answer as a list separated by commas, such as "1, 2, 3, 4, 5". If there are no such [tex]n[/tex], write 0. If there is only one such [tex]n[/tex] give that value.)

[Guest]

n+4 is a multiple of 3 if and only if n+1 is (because they differ by 3).

So n, n+2, n+4 contains a multiple of 3 if and only if n, n+2, n+1 contains a multiple of 3, but the latter is a list of 3 consecutive numbers, so must contain a multiple of 3.

So one of the primes is 3, the only solution is n=3.

Hope this helped,

R. Baber.