An Algorithm to Find Sums of Powers of Consecutive Primes

TL;DR

This paper introduces an algorithm to efficiently enumerate integers up to x that are sums of consecutive k-th powers of primes, providing asymptotic bounds and extending previous work on sums of prime squares.

Contribution

The paper presents a novel algorithm for counting sums of consecutive prime powers and establishes their asymptotic bounds, extending prior research on prime squares.

Findings

Asymptotic bound on the number of such integers proportional to x^{2/(k+1)}

Development of a fast counting algorithm for these integers

Extension of previous work from prime squares to higher powers

Abstract

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant times $$ c_k \frac{ x^{2/(k+1)} }{ (\log x)^{2k/(k+1)} }, $$ where $c_k$ is a constant depending solely on $k$, roughly $k^2$ in magnitude. This also bounds the asymptotic running time of our algorithm. We also give a lower bound of the same order of magnitude, and a very fast algorithm that counts such $n$. Our work extends the previous work by Tongsomporn, Wananiyakul, and Steuding (2022) who examined sums of squares of consecutive primes.