Month: March 2024

I’m continuing my evaluation of Khashin’s Frobenius Primality Test from his 2013 preprint at the URL https://arxiv.org/abs/1307.7920. To make it easier to assess the accuracy of this test, I assume that in a first round, one conducts a Fermat base 2 test, which will detect all composites that are not base 2 pseudoprimes. In round… Continue reading Continued Evaluation of a Frobenius Primality Test

With respect to the enhanced Lucas test for n with parameters P and Q, using congruences for both of the Lucas sequences U_n and V_n,I’ve come to the realization that the Lucas congruences are equivalent to a specific test inspired by the Frobenius endomorphism in the ring Z/nZ[sqrt(D)] where D=P^2-4Q.Specifically, and assuming P and Q… Continue reading Equivalence of Enhanced Lucas Test with a Frobenius Test

Certainly! Here’s a concise summary of the equivalence between an enhanced Lucas test and a Frobenius test: The equivalence between an enhanced Lucas test, which includes congruences for both (U_n) and (V_n) sequences, and a Frobenius test, which checks if (x^n = \text{conjugate}(x)) for a specific choice of (x), lies in their shared foundation in… Continue reading Enhanced Lucas Test, and a Frobenius Test by Chat-GPT

The exploration of primality tests is a foundational pillar in the realm of computational number theory, with wide-ranging applications from cryptography to the distribution of prime numbers. Among the myriad of tests developed over the years, the Lucas sequences and Frobenius tests stand out for their unique approaches and theoretical underpinnings. Our recent investigation has… Continue reading Exploring a Lucas-Frobenius Tests Connection by Chat-GPT

A promising primality test is Sergei Khashin’s Frobenius test, as described in a 2013 arxiv preprint of his, Counterexamples for Frobenius primality test, available from the url https://arxiv.org/abs/1307.7920. The idea of the Frobenius test originated with Jon Grantham, see for example his 1998 article A Probable Prime Test With High Confidence available from the url… Continue reading Evaluation of a Frobenius Primality Test Using Pseudoprimes Tables

Given an odd number n>=3, non-square and an odd number c such that Jacobi(c,n)=-1 and 1<c<n/2,Khashin’s Frobenius primality test checks whether the congruence (1+sqrt(c))^n == 1-sqrt(c) (modulo n)holds. This congruence holds whenever n is a prime number, thanks to properties of the Frobeniusautomorphism map x |-> x^p where p is prime and x is in… Continue reading False positive rates in Khashin’s Frobenius primality test