joshipura wrote: > >I am not a mathematician - but can understand prime numbers, >and even the hypothesis under discussion. > >I wanted to tell to my children (who also know about prime >numbers) about this development. Here is my script: > >"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 if p is a prime number, within >p + 70,000,000 there is another prime number q, no matter >how large p is.
Let d = 70,000,000.
It's not true that for all primes p there is a prime q with p < q <= p + d.
For example, let p be the largest prime less than (d + 1)! + 2 and let q be the least prime greater than p. Then q > p + d, so there are no primes in the range p + 1 to p + d inclusive.
What was proved is that there are infinitely many primes pairs p,q with p < q <= p + d.