Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdos on Consecutive Integers

Abstract

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E 2 -values; i.e., values that are products of exactly two primes.We use that result to prove that there are inifinitely many integers x that simultaneously satisfy ω(x) = ω(x + 1) = 4, Ω(x) = Ω(x + 1) = 5, and d(x) = d(x + 1) = 24.Here, ω(x), Ω(x), d(x) represent the number of prime divisors of x, the number of prime power divisors of x, and the number of divisors of x, respectively.We also prove similar theorems where x + 1 is replaced by x + b for an arbitrary positive integer b.Our results sharpen earlier work of Heath-Brown, Pinner, and Schlage-Puchta.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF
• CiteSeer X (The Pennsylvania State University) - View - PDF