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 $$\omega(x)=\omega(x+1)=4, \Omega(x)=\Omega(x+1)=5, \text{and} d(x)=d(x+1)=24.$$ Here, $\omega(x), \Omega(x), d(x)$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2008
- DOI:
- 10.48550/arXiv.0803.2636
- arXiv:
- arXiv:0803.2636
- Bibcode:
- 2008arXiv0803.2636G
- Keywords:
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- Mathematics - Number Theory;
- 11N37