Daniel Goldston

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Daniel Alan Goldston (born January 4, 1954 in Oakland, California) is an American mathematician who specializes in number theory. He is currently a professor of mathematics at San Jose State University.

Goldston is best known for the following result that he, János Pintz, and Cem Yildirim proved in 2005 [1]:

$\liminf_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0$

where $p_n\$ denotes the nth prime number. In other words, for every $c>0\$ , there exist infinitely many pairs of consecutive primes $p_n\$ and $p_{n+1}\$ which are closer to each other than the average distance between consecutive primes by a factor of $c\$ , i.e., $p_{n+1}-p_n<c\log p_n\$ .

In fact, if they assume the Elliott-Halberstam conjecture, then they can also show that primes within 16 of each other occur infinitely often, which is nearly the twin prime conjecture.