The project started on November 1st 2008 and will last until October31, 2013. Our main fields of research are the following:

- To investigate the appearance of small gaps between consecutive primes approximating the twin prime conjecture as much as possible.
- To investigate small differences and possible configurations in almost primes, in particular for consecutive semiprimes (i.e. numbers with exactly two prime factors).
- To prove a suitable analogue of the Green-Tao theorem, that is the existence of arbitrarily long finite arithmetic progressions of primes - assuming suitable uniform distribution of primes in arithmetic progressions with large moduli (i.e. a hypothetical improvement of the Bombieri-Vinogradov theorem) - such that each prime p in the progression is at the same bounded difference from the prime p' following p. That is to find arbitrary long arithmetic progressions of pairs of consecutive primes, where the distance is the same bounded quantity in case of all pairs.
- To consider the same problem unconditionally when primes are substituted by semiprimes.
- To investigate approximations of some famous additive problems on primes, as the Goldbach or Hardy-Littlewood problems, with special emphasis on the density and distribution of exceptional numbers which do not satisfy the corresponding conjecture.

If you are interested in , please contact pintz@renyi.hu

## Staff:

- Principal Investigator: Prof. János Pintz
- Prof. Endre Szemerédi
- Prof. Imre Ruzsa
- Dr. András Bíró
- Dr. Máté Matolcsi
- Dr. Szilárd Révész

## Visiting researchers:

- Prof. Roger Baker
- Prof. Francois Hennecart