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… Expand

The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the application of a theorem of Watt (1995)… Expand

We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J/ n{l,...,«} is O(l/log«) c »), cn->-oo. This improves previous… Expand

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ⩽ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of… Expand

Assume the Generalized Riemann Hypothesis and suppose thatHlog−6x→t8. Then we prove that all even integers in any interval of the form (x, x, +H) butO(H1/2log3x) exceptions are a sum of two primes.

Linnik considered about 70 years ago the following approximation to the binary Goldbach problem. Is it possible to give a fixed integer K such that every sufficiently large even integer could be… Expand