Average liar count for degree-2 Frobenius pseudoprimes

In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham, generalizing arguments of Erd\H{o}s and Pomerance, and Monier. These bounds are provided for both Jacobi symbol plus and minus cases, providing evidence for the existence of several challenge pseudoprimes.Comment: 19 pages, published in Mathematics of Computation, revised version fixes typos and made a minor correction to the proof of Lemma 18 (result remains unchanged

SOME REMARKS ON LUCAS PSEUDOPRIMES

We present a way of viewing Lucas pseudoprimes, Euler-Lucas pseudoprimes and strong Lucas pseudoprimes in the context of group schemes. This enables us to treat the Lucas pseudoprimalities in parallel to establish pseudoprimes, Euler pseudoprimes and strong pseudoprimes

Primality tests, linear recurrent sequences and the Pell equation

We study new primality tests based on linear recurrent sequences of degree two exploiting a matricial approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce a new specific primality test, which we will call generalized Pell test. We perform some numerical computations on the new primality tests and, for the generalized Pell test, we do not any pseudoprime up to $10^{10}$

Primality tests, linear recurrent sequences and the Pell equation

4We study new primality tests based on linear recurrent sequences of degree two exploiting a matrix approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce two new specific primality tests, which we will call generalized Lucas test and generalized Pell test. We perform some numerical computations on the new primality tests and we do not find any pseudoprime up to 238. Moreover, we combined the generalized Lucas test with the Fermat test up to 264 and we did not find any composite number that passes the test. We get the same result using the generalized Pell test.partially_openembargoed_20220207Bazzanella, Danilo; Di Scala, Antonio; Dutto, Simone; Murru, NadirBazzanella, Danilo; Di Scala, Antonio; Dutto, Simone; Murru, Nadi

Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.Comment: 25 page