>OK. So here goes changed script for review: > >"For years mathematicians are struggling to prove that they >will always find larger and larger cases of p where p and p+2 >both are primes. > >Someone recently proved that >** >there are as many prime numbers p and q less than 70,000,000 >apart as you want >**
Yes, In a sense.
The phrase "as many as you want" can be interpreted as "infinitely many" (provided you always want more and more).
>So, now mathematicians will work on finding what types of >p's this 70 million is negotiable to smaller numbers, >eventually going down to 2."
It's not likely that they'll discover a classification for the primes which have nearby successor primes.
They simply need to show, for a given integer d >= 2, that there are infinitely prime pairs p,q with p < q <= p + d.
But yes, the goal is to keep reducing the value of d for which they can prove the existence of infinitely many such prime pairs, all the way down to d = 2, if possible.