On a Combined Primality Test

Abstract

A new combined primality test for natural numbers consisting of the Lucas test and checking Fermat’s condition ${{2}^{{n - 1}}} \equiv 1\,(\bmod n)$ is studied. We call this procedure an L2 test. Composite numbers passing the L2 test are called L2 pseudoprimes. We give a description of a new effective algorithm for searching for L2 pseudoprimes; using it, we show that there is no L2 pseudoprime $n$ of a special form $n \equiv \pm 2\,(\bmod 5)$ less $B = {{10}^{{23}}}$ (this border has been reached to date and is constantly increasing). Thus, the L2 test is a deterministic test that allows one to determine the primality of natural numbers $n \equiv \pm 2\,(\bmod 5)$ at least up to ${{10}^{{23}}}$ using only two iterations, each of which has a computational complexity of $O({{\ln }^{3}}n)$.