Are there an infinite number of sizes of gaps between primes? let $p_n$ be the nth prime number. Let $g_n = p_{n+1} – p_n$ (i.e. size of gaps between consecutive primes). As $p_n$ goes to infinity, does $g_n$ go to infinity also?
[Math] Are there infinite number of sizes of gaps between primes
number theory
Best Answer
You can easily find as long a string of composites as you wish, so the gaps between primes can be arbitrarily large, so must have infinitely many different values.
Consecutive composite numbers
But that does not mean the size of the gap goes to infinity. In fact it's less than 70 million infinitely often.
https://en.wikipedia.org/wiki/Yitang_Zhang
As @DunstanLevenstein comments. 70 million was the bound in Zhang's revolutionary paper. It's since been reduced to 246.
It's thought that in fact there are infinitely many twin primes, so the conjecture is that the bound is actually 2.