János Pintz
János Pintz (born 20 December 1950 in Budapest) is a Hungarian mathematician working in analytic number theory. He is a fellow of the Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences. In 2014, he received the Cole Prize. Mathematical results. Pintz is best known for proving in 2005 (with Daniel Goldston and Cem Yıldırım) that where formula_2 denotes the "n" prime number. In other words, for every ε > 0, there exist infinitely many pairs of consecutive primes "p" and "p" that are closer to each other than the average distance between  consecutive primes by a factor of ε, i.e., "p" − "p" < ε log "p". This result was originally reported in 2003 by Daniel Goldston and Cem Yıldırım but was later retracted. Pintz joined the team and completed the proof in 2005. Later they improved this to showing that "p" − "p" < ε(log log "n") occurs infinitely often. Further, if one assumes the Elliott–Halberstam conjecture, then one can also show that primes within 16 of each other occur infinitely often, which is nearly the twin prime conjecture. Additionally,