An Algorithm to Find Sums of Powers of Consecutive Primes

DOI10.5281/ZENODO.10450932arXiv2204.10930MaRDI QIDQ6507812

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Abstract: We present and analyze an algorithm to enumerate all integers $nlex$ 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 ${\displaystyle }$ c_k frac{ x^{2/(k+1)} }{ (log x)^{2k/(k+1)} }, ${\displaystyle }$ 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.

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