Dense admissible sequences (Q2723540)

A sequence of integers \(b_1 < b_2 < \cdots < b_k \) in an interval of length \(x\) is called admissible if for each prime \(p\), there is a residue class modulo \(p\) which contains none of the \(b_i \). For such an admissible sequence Hardy and Littlewood's prime \(k\)-tuples conjecture states that there exist infinitely many integers \(n\) for which \(n+b_1 , n+b_2 , \ldots , n+ b_k \) are prime. Denoting the maximum number of elements in an admissible sequence in an interval of length \(x\) by \(\rho^{*}(x)\), and defining \(\rho(x) = \limsup_{y\to\infty}\pi(y+x)- \pi(y)\), the prime \(k\)-tuples conjecture implies that \(\rho^{*}(x)=\rho(x)\). \textit{D. Hensley} and \textit{I. Richards} [Acta Arith. 25, 375-391 (1974; Zbl 0285.10004)] proved that for sufficiently large \(x\), \(\rho^{*}(x) > \pi(x)\), thereby showing that the prime \(k\)-tuples conjecture is incompatible with Hardy and Littlewood's other conjecture that \(\pi(x+y)-\pi(y) \leq \pi(x)\). NEWLINENEWLINENEWLINEIn this paper the authors find values of \(x\) satisfying \(\rho^{*}(x) > 2\pi(x/2)\), so that the widely believed prime \(k\)-tuples conjecture is also incompatible with Erdős's weaker conjecture \(\pi(x+y)-\pi(y)\leq 2\pi(x/2)\). Moreover the known bounds on the set of \(x\) satisfying \(\rho^{*}(x)\leq \pi(x)\) are increased, and smaller values of \(x\) for which \(\rho^{*}(x) > \pi(x)\) are found. The authors give an algorithm for the computation and provide some tables of their results.