On the distribution of primes in the alternating sums of concecutive primes

Romeo Meštrović

paper

On the distribution of primes in the alternating sums of concecutive primes

arXiv:1805.11657

Abstract

Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_{n})$ whose $n$ th term is defined as $S_{n} = \sum_{k = 1}^{2 n} p_{k}$ , where $p_{k}$ is the $k$ th prime. Some heuristic arguments and the numerical evidence lead to the conjecture that the primes are distributed among sequence $(S_{n})$ in the same way that they are distributed among positive integers. More precisely, Conjecture 3.3 in [8] asserts that $π_{n} \sim \frac{n}{l o g n}$ as $n \to \infty$ , where $π_{n}$ denotes the number of primes in the set ${S_{1}, S_{2}, \dots, S_{n}}$ . Motivated by this, here we consider the distribution of primes in aletrnating sums of first $2 n$ primes, i.e., in the sequences $(A_{n})$ and $(T_{n})$ defined by $A_{n} := \sum_{i = 1}^{2 n} (- 1)^{i} p_{i}$ and $T_{n} := A_{n} - 2 = \sum_{i = 2}^{2 n} (- 1)^{i} p_{i}$ ( $n = 1, 2, \dots$ ). Heuristic arguments and computational results suggest the conjecture that (Conjecture 2.5) $π_{(A_{k})} (A_{n}) \sim π_{(T_{k})} (T_{n}) \sim \frac{2 n}{lo g n} as n \to \infty,$ where $π_{(A_{k})} (A_{n})$ (respectively, $π_{(T_{k})} (T_{n})$ ) denotes the number of primes in the set ${A_{1}, A_{2}, \dots, A_{n}}$ (respectively, ${T_{1}, T_{2}, \dots, T_{n}}$ ). Under Conjecture 2.5 and Pillai's conjecture, we establish two results concerning the expressions for the $k$ th prime in the sequences $(A_{n})$ and $(T_{n})$ . Furthermore, we propose some other related conjectures and we deduce some their consequences.

9 pages, one table, no figures