On the number of solutions of \(p+h=P_ r\)
Let \(x\) be a sufficiently large positive number, h(\(\neq 0)\) a fixed even number, \(p,p_ 1,...,p_ r\) primes and \(P_ r\) an almost prime with at most r prime factors counted with multiplicity. The work to search for the lower bound for \({\#}\{p: p\leq x\), \(p+h=P_ r\}\) as well as possible is closely connected with the well known Twin Prime Conjecture. In all papers published up to date the order of magnitude for the lower bound is at most \(x \ln^{-2}x\), much smaller than the expected order \(x \ln^{-2}x(\ln \ln x)^{r-1}\). In this paper we show the following: Theorem 1. For \(r\geq 3\), \({\#}\{p: p\leq x\), \(p+h=P_ r\}\gg_{h,r}^ x \ln^{-2}(\ln \ln x)^{r-2};\) Theorem 2. For \(r\geq 3\), \({\#}\{p: p\leq x\), \(p+h=p_ 1...p_{r-2}\) or \(p_ 1...p_{r-1}\) or \(p_ 1...p_ r\}\gg_{h,r}^ x \)ln\({}^{- 2}x(\ln \ln x)^{r-2}.\) Theorem 3. \({\#}\{p: p\leq x\), \(p+h=p_ 1p_ 2\) or \(p_ 1p_ 2p_ 3\}\gg_ h\;x \ln^{-2}x \ln \ln x;\) and miss the presumably correct order of magnitude only by a factor of \(\ln\ln x\).
scientific article; zbMATH DE number 4077365