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

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

2010-07-06

arxiv.org/abs/1007.0929v1

Mathematics

Number Theory

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

An $\tilde{O}(\log^2(N))$ time primality test for Generalized Cullen Numbers does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with An $\tilde{O}(\log^2(N))$ time primality test for Generalized Cullen Numbers, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An $\tilde{O}(\log^2(N))$ time primality test for Generalized Cullen Numbers will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-216943