Combined Sieve Algorithm for Prime Gaps

TL;DR

The paper introduces a novel Combined Sieve algorithm that significantly accelerates the search for large prime gaps by leveraging a space-time tradeoff and a new modular equation solution, achieving up to 10,000x speedup.

Contribution

It presents a new combined sieve method that outperforms traditional sieves in speed and efficiency for large interval prime gap searches.

Findings

Achieved up to 10,000x speedup over traditional sieve methods.

Discovered two top-10 record prime gaps during testing.

Accelerated prime gap searches by 30-70% using higher sieving limits.

Abstract

A new Combined Sieve algorithm is presented with cost proportional to the number of enumerated factors over a series of intervals. This algorithm achieves a significant speedup, over a traditional sieve, when handling many ([10^4, 10^7]) intervals concurrently. The speedup comes from a space-time tradeoff and a novel solution to a modular equation. In real world tests, this new algorithm regularly runs 10,000x faster. This faster sieve paired with higher sieving limits eliminates more composites and accelerates the search for large prime gaps by 30-70%. During the development and testing of this new algorithm, two top-10 record merit prime gaps were discovered.