János Pintz


János Pintz 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 that
where denotes the nth prime number. In other words, for every ε > 0, there exist infinitely many pairs of consecutive primes pn and pn+1 that are closer to each other than the average distance between consecutive primes by a factor of ε, i.e., pn+1 − pn < ε log pn. 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 pn+1 − pn < ε2 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,