Cem Yıldırım
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Cem Yalçın Yıldırım is a Turkish mathematician who specializes in number theory. He obtained his PhD from the University of Toronto in 1990. His advisor was John Friedlander. He is currently a faculty member at Boğaziçi University in Istanbul, Turkey.
In 2005([1]), with Dan Goldston and János Pintz, he proved, that for any positive number ε there exist primes p and p′ such that the difference between p and p′ is smaller than ε log p.
Formally;
where 
denotes the nth prime number. In other words, for every 
, there exist infinitely many pairs of consecutive primes and 
which are closer to each other than the average distance between consecutive primes by a factor of 
, i.e., 
.
This result was originally reported in 2003 by Dan Goldston and Cem Yıldırım but was later retracted[2][3]. Then Janos Pintz joined the team and they completed the proof in 2005.
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.
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