# 3, 5, and 7 are the only Prime Triplets

A classic unsolved problem in number theory asks if there are infinitely many pairs of `twin primes', pairs of primes separated by 2, such as 3 and 5, 11 and 13, or 71 and 73. Prove that the only prime triple (i.e. three primes, each 2 from the next) is 3, 5, 7.

© BrainMass Inc. brainmass.com October 10, 2019, 5:52 am ad1c9bdddfhttps://brainmass.com/math/number-theory/only-prime-triplets-520657

#### Solution Preview

To prove that 3 5 and 11 are the only Triplets.

Given: Many twin primes exist.

Any positive integer can be expressed in the following form

3n, 3n+1, 3n+2

Conside 3n:

Let 3n be the first prime P

Let 3n+2 be the second prime p+2 (twin prime of p)

let 3n+4 be the third prime p+4 (prime triple of p, p+2 and p+4)

for n =1, 3n, 3n+2, 3n+4 would give you ...

#### Solution Summary

This solution evaluates all the prime numbers and prove that every third number (odd number) is not a prime except 7. In that process prove that Prime twins exist.