An Algorithm to Find Sums of Powers of Consecutive Primes (2204.10930v2)

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.

• The paper introduces optimal enumeration and efficient counting algorithms that compute sums of powers of consecutive primes with linear complexity relative to the number of sums.
• The authors derive tight asymptotic bounds, showing that the upper bound involves a coefficient approximately equal to k², which is critical for understanding the density of these sums.
• Empirical validation supports the theoretical predictions, underscoring potential applications in cryptography and further computational number theory research.

An Overview of "Algorithms and Bounds on the Sums of Powers of Consecutive Primes"

The paper "Algorithms and Bounds on the Sums of Powers of Consecutive Primes" presents a thorough exploration into the field of number theory and computational mathematics, specifically focusing on enumeration and counting algorithms related to the sums of powers of consecutive prime numbers. The authors, Cathal O'Sullivan, Jonathan P. Sorenson, and Aryn Stahl, extend previous work by developing more general algorithms applicable for various powers $k>1$.

Theoretical Contributions

The paper establishes asymptotic bounds on the count of integers $n$ that can be expressed as sums of $k$-th powers of consecutive primes. The upper bound is shown to be: $c_k \frac{x^{2/(k+1)}}{(\log x)^{2k/(k+1)}}$ where $c_k$ is approximately $k^2$. The authors also present a lower bound as: $$\frac{(k+1)^2}{2} \frac{x^{2/(k+1)}}{(\log x)^{2k/(k+1)}}.$$ These bounds provide critical insights into the density and frequency of such numbers, offering a window into the larger structure of prime numbers and their powers.

Algorithmic Approach

The paper describes two algorithms: one for enumeration and one for counting. The enumeration algorithm is optimal in terms of its operation count and explicitly constructs all integers $n \le x$ with minimal computational overhead. The counting algorithm, on the other hand, leverages the efficiency of finding primes up to $x^{1/k}$ and efficiently computes the total number $s(x)$.

The authors achieve an optimal enumeration algorithm with linear complexity relative to the number of sums, $s(x)$. The counting algorithm improves computational efficiency by focusing on the prime list computation, which is the primary time constraint.

Empirical Validation

To validate their theoretical results, the authors conduct computational experiments, computing the sums for various values of $k$ and $x$. These results empirically support the theoretical bounds derived. The alignment between empirical data and theoretical predictions demonstrates the robustness and accuracy of the proposed bounds and algorithms.

Implications and Future Directions

From a theoretical standpoint, these algorithms enhance the understanding of the relationships between sums of powers of primes. Practically, such insights can be pivotal in cryptographic applications where prime number generation and computation efficiencies are paramount.

The authors propose several future research directions, including parallelizing the algorithms to handle larger powers and establishing explicit constants for their upper bounds. Additionally, the search for integers with multiple representations as sums of powers of consecutive primes beyond the trivial case $k=2$ remains open and poses an intriguing question for further studies.

Conclusion

This paper delivers significant contributions to computational number theory, offering robust algorithms accompanied by strong theoretical bounds related to sums of powers of consecutive primes. Through careful analysis and empirical validation, it sets a foundational framework for further exploring this niche yet fundamental aspect of number theory in both theoretical and practical domains.