Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers

Details Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers

2008-03-18

arxiv.org/abs/0803.2636v1

Mathematics

Number Theory

Scientific paper

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.

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