Prime triplet

A set of three prime numbers which form an arithmetic sequence with common difference two is called a prime triplet.

Discussion

An example of a prime triplet is $\{3,5,7\}$ .

$\{3,5,7\}$ turns out to be the only prime triplet. This is because any set $\{n,n+2,n+4\} \pmod 3$ becomes $\{0,2,1\}$ , $\{2,1,0\}$ , or $\{1,0,2\}$ . Therefore in every triplet there exists one number that is divisible by $3$ . The only prime number divisible by $3$ is $3$ itself, so the only triplets possible are $\{1,3,5\}$ and $\{3,5,7\}$ . Since $1$ is not a prime, $\{3,5,7\}$ is the only prime triplet.

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Prime triplet

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