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

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 $$\omega (x)=\omega (x+1)=4,\mathrm{\Omega }(x)=\mathrm{\Omega }(x+1)=5,\text{and}d(x)=d(x+1)=24.$$ Here, $\omega(x), \Omega(x), d(x)$representthenumberofprimedivisorsof$x$, 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.