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

TL;DR

This paper investigates the distribution of primes within sequences formed by alternating sums of the first 2n primes, proposing conjectures that these primes are distributed similarly to natural numbers, supported by heuristic and computational evidence.

Contribution

It introduces new conjectures about prime distribution in alternating prime sums and derives results under these conjectures, extending previous work on prime sequences.

Findings

Heuristic and computational evidence support the conjecture of prime distribution in the sequences.

Under the conjectures, formulas for the nth prime in the sequences are established.

Proposes related conjectures and explores their implications.

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 $S_n=\sum_{k=1}^{2n}p_k$, 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 $\pi_n\sim \frac{n}{\log n}$ as $n\to \infty$, where $\pi_n$ denotes the number of primes in the set $\{S_1,S_2,\ldots, S_n\}$. Motivated by this, here we consider the distribution of primes in aletrnating sums of first $2n$ primes, i.e., in the sequences $(A_n)$ and $(T_n)$ defined by $A_n:=\sum_{i=1}^{2n}(-1)^ip_i$ and $T_n:=A_n-2=\sum_{i=2}^{2n}(-1)^ip_i$ ($n=1,2,\ldots$). Heuristic arguments and computational results suggest the conjecture that (Conjecture 2.5) $$ \pi_{(A_k)}(A_n)\sim \pi_{(T_k)}(T_n)\sim \frac{2n}{\log n} \quad {\rm as}\,\, n\to \infty, $$ where $\pi_{(A_k)}(A_n)$ (respectively, $\pi_{(T_k)}(T_n)$) denotes the number of primes in the set $\{A_1,A_2,\ldots, A_n\}$(respectively, $\{T_1,T_2,\ldots, T_n\}$). 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.