An $\tilde{O}(\log^2(N))$ time primality test for Generalized Cullen Numbers

TL;DR

This paper introduces a new primality test for Generalized Cullen Numbers that is faster and more reliable than existing tests, running in nearly logarithmic squared time and showing very few pseudoprimes.

Contribution

It presents a novel primality test for Generalized Cullen Numbers with improved efficiency and fewer pseudoprimes, including a quasi-deterministic method with $ ilde{O}( ext{log}^2(N))$ complexity.

Findings

The first test is more efficient than Fermat and Miller-Rabin tests.

Very few pseudoprimes (only 4) were found among tested numbers.

The second test is pseudoprime-free and runs in $ ilde{O}( ext{log}^2(N))$ time.

Abstract

Generalized Cullen Numbers are positive integers of the form $C_b(n):=nb^n+1$. In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is based in the following property of primes from this family: $n^{b^{n}}\equiv (-1)^{b}$ (mod $nb^n+1$). It is stronger and has less computational cost than Fermat's test (for bases $b$ and $n$) and than Miller-Rabin's test (for base $n$). Pseudoprimes for this new test seem to be very scarce, only 4 pseudoprimes have been found among the many millions of Generalized Cullen Numbers tested. We also present a second, more demanding, test for wich no pseudoprimes have been found. This test leads to a "quasi-deterministic" test, running in $\tilde{O}(\log^2(N))$ time, which might be very useful in the search of Generalized Cullen Primes.