Question

Problem

A prime trio is a collection of three prime numbers
`\{p,q,r\}` in arithemtic progression, with common difference
`q-p=r-q`. For example, the prime trio
`\{3,5,7\}`has common difference 2.
(a) Find a prime trio with common difference 50.
(b) Find two prime trios with common difference 60.
(c) Prove that there is no prime trio with common difference 70.

Answer 1

(a) 50:

3 53 103

(b) 60:

7 67 127

11 71 131

Explanation:

OK, I think you can prove this with the theorem that any prime greater than 3 is 6n-1 or 6n+1, for integer n. In other words, any prime is near a multiple of 6; it is either one less or one more. The proof for this is very easy, look it up.

Given this fact, if two primes (p,q) are 70 a part, this means the first one has to be a case of 6n+1 and the second one has to be a case of 6n-1. Why? Because the nearest multiple of 6 is 72 and with +1 on one end and -1 on the other end, you can reach a "gap" of 70.

Now for the next two primes (q,r) the same must hold. Only, it can't because this requires q to be both 6n-1 and 6n+1.