>I must say that some of the popular exposition of this discovery of Zhang has been to allow the impression that there is a limit to the size of prime

>gaps, a proposition that puzzled me for a while, but which would have never occured to a Mathematician worth of the name due to its implicit dismissal of established theory.

There is no limit to the maximum gap, consider n!+2, n!+3, ... n!+n, these are all composite for large n, but it is trivial to show that up to a point x there is a maximal gap.

> All that Zhang seems to have assured us, and this is the comforting news, is that no how far along the number line we go, that at some stage we will find a prime gap of less than about 70 million.

That is correct.

> In fact, it seems to be to imply that for some value less than the "Zhang Number" which is about 70 million, there is an infinite number of

> prime gaps, which fact gives some hope of proving the twin primes is infinite. But, the obvious gets no Brownie points, which is

> not to say that Zhang should not get the credit he deserves.

To have proved the first fixed limit is an amazing result and will probably prepare the way for a long sequence of results improving the result.