On the several differences between primes (Q1429328)

For a given positive integer \(k\), the number of primes \(p\leq x\) such that \(p+2k\) is also prime is denoted by \(\pi_{2,2k}(x)\), or by \(\pi^*_{2,2k}(x)\) if \(p\) and \(p+2k\) are consecutive primes (this is, of course, important only if \(k\geq 3\)). The authors have calculated \(\pi_{2,2}(x)\), \(\pi^*_{2,4}(x)\), and \(\pi^*_{2,6}(x)\) for various values of \(x\leq 5\times 10^{10}\). The results are in good agreement with conjectures involving logarithmic integrals. As with the known result for \(k=1\), the sums of reciprocals of such primes, when \(k=2\) or \(3\), are finite, but whether or not there are infinitely many such pairs is still unknown. \{On page 43, in the proof of Theorem 2.5, the reference should be to Theorem 2.4\}.