A question on Zhang's result on prime gaps

Question

I'd like to know which is the right way to mention the result that Yitang Zhang obtained in his paper "Bounded gaps between primes".

In some places it is said that Zhang proved that there are infinitely many pairs of prime numbers which differ by $70,000,000$ or less, whereas in other places it is said that he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$.

In his paper, Zhang mentions that it is conjectured that

$$\liminf_{n\to\infty}(p_{n+1}-p_n)=2,$$

which is the Twin Prime Conjecture. Besides, the abstract states that he proves that

$$\liminf_{n\to\infty}(p_{n+1}-p_n)<7\times 10^7.$$

So, I think that the right thing is to say that Zhang proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$.

Any comments?

Answer 1

I think that the right way to formulate this is the following:

There is a number $L < 70000000$ such that there are infinitely many pairs of primes which differ by $L$.

Answer 2

If there are infinitely many pairs of primes which differ by at most $L$, then there must be some $\ell\le L$ such that there are infinitely many pairs of primes which differ by exactly $\ell$ (since if not, the total number of pairs of primes differing by at most $L$ is at most $L$ times some finite number which is still finite). So the formulations are equivalent.

Another equivalent formulation is that there is some $\ell\le L$ for which there are infinitely many pairs of consecutive primes which differ by exactly $\ell$.

(Expanding on the first paragraph as requested:)

As an example, consider the current bound: there are infinitely many prime gaps of length at most 246. Since any such gap must be a positive even integer, the only choices for such a gap are 2, 4, …, 244, and 246. If there were only finitely many gaps of length 2, and only finitely many gaps of length 4, and so on up to 246, then there would have to be only finitely many gaps of length at most 246. Hence one of these must be wrong, and there is some $2 \le n \le 246$ such that there are infinitely many gaps of length n.