Primes in tuples I


We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, \[ \liminf_{n\to \infty} \frac{p_{n+1}-p_n}{\log p_n} =0 .\] We will quantify this result further in a later paper.


Daniel A. Goldston

Department of Mathematics
San Jose State University
One Washington Square
San Jose, CA 95192-0130
United States

János Pintz

Alfred Rényi Institute of Mathematics
Hungarian Academy of Sciences
P.O. Box 127
1364 Budapest

Cem Y. Yíldírím

Feza Gürsey Enstitüsü
Kuleli Mahallesi, Şekip Ayhan Özışık Caddesi 44
34684 Çengelköy, İstanbul