Deterministic methods to find primes

TL;DR

This paper explores deterministic methods for finding primes efficiently, proposing strategies that could improve current bounds and allow partial results like determining the parity of the number of primes in an interval.

Contribution

It suggests a new approach to determine in sub-$O(N^{1/2})$ time whether an interval contains a prime, potentially improving existing deterministic algorithms.

Findings

Proposes a strategy to check for primes in an interval in $O(N^{1/2-c})$ time.

Shows how to determine the parity of the number of primes in an interval efficiently.

Provides partial results under the new approach, advancing deterministic prime search methods.

Abstract

Given a large positive integer $N$, how quickly can one construct a prime number larger than $N$ (or between $N$ and 2N)? Using probabilistic methods, one can obtain a prime number in time at most $\log^{O(1)} N$ with high probability by selecting numbers between $N$ and 2N at random and testing each one in turn for primality until a prime is discovered. However, if one seeks a deterministic method, then the problem is much more difficult, unless one assumes some unproven conjectures in number theory; brute force methods give a $O(N^{1+o(1)})$ algorithm, and the best unconditional algorithm, due to Odlyzko, has a run time of $O(N^{1/2 + o(1)})$. In this paper we discuss an approach that may improve upon the $O(N^{1/2+o(1)})$ bound, by suggesting a strategy to determine in time $O(N^{1/2-c})$ for some $c>0$ whether a given interval in $[N,2N]$ contains a prime. While this strategy has not been fully implemented, it can be used to establish partial results, such as being able to determine the \emph{parity} of the number of primes in a given interval in $[N,2N]$ in time $O(N^{1/2-c})$.