Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers
| dc.creator | Goldston, D. A. | |
| dc.creator | Graham, S. W. | |
| dc.creator | Pintz, J. | |
| dc.creator | Yildirim, C. Y. | |
| dc.date | 2008-03-18 | |
| dc.date.accessioned | 2026-07-07T09:27:22Z | |
| dc.date.available | 2026-07-07T09:27:22Z | |
| dc.description | 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 $$ω(x)=ω(x+1)=4, Ω(x)=Ω(x+1)=5, \text{and} d(x)=d(x+1)=24.$$ 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. | |
| dc.identifier | https://arxiv.org/abs/0803.2636 | |
| dc.identifier | http://arxiv.org/abs/0803.2636 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/157075 | |
| dc.subject | Number Theory | |
| dc.subject | 11N37 | |
| dc.title | Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers | |
| dc.type | text |