# Small gaps between products of two primes

```@article{Goldston2009SmallGB,
title={Small gaps between products of two primes},
author={D. A. Goldston and Sidney W. Graham and Janos Pintz and C. Y. Yildirim},
journal={Proceedings of the London Mathematical Society},
year={2009},
volume={98}
}```
• Published 21 September 2006
• Mathematics
• Proceedings of the London Mathematical Society
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 6. More generally, we prove that if ν is any positive integer, then (qn+ν − qn) ⩽ ν eν − γ (1+o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms.
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## References

SHOWING 1-10 OF 58 REFERENCES
SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES
• Mathematics
• 2005
Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some
Primes in tuples I
• Mathematics
• 2009
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
The equation ?( n) = ?( n + 1)
Our argument will be similar to the one given by D. R. Heath-Brown [1] for consecutive integers with the same number of divisors. The method given there can be applied to a variety of similar
ON THE REPRESENTATION OF A LARGER EVEN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF AT MOST TWO PRIMES
In this paper we shall prove that every sufficiently large even integer is a sum of a prime and a product of at most 2 primes. The method used is simple without any complicated numerical calculations.
The Distribution of Values of the Divisor Function d(n)
• Mathematics
• 1952
1. THROUGHOUT this note the letters p, q will be reserved for primes, and pV will denote the vth prime; cr, q,,... are to stand for absolute positive constants. Let d(n) denote, as usual, the number
Limitations to the equi-distribution of primes I
• Mathematics
• 1989
In an earlier paper FG] we showed that the expected asymptotic formula (x; q; a) (x)==(q) does not hold uniformly in the range q < x= log N x, for any xed N > 0. There are several reasons to suspect
SOME PROBLEMS ON NUMBER THEORY
A k = max(p,+ t p,), k Ak for all large k. A well known theorem of Polya and Stormer states that if u > uo(k) then u(u + 1) always contains a prime factor greater than k, thusf(k) can be determined
Basic Analytic Number Theory
I. Integer Points.- 1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results.- 2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums.- 3. Theorems