On the Baillie PSW Conjecture

Abstract

The Baillie PSW conjecture was formulated in 1980 and was named after its authors Baillie, Pomerance, Selfridge, and Wagstaff, Jr. The conjecture is related to the problem of the existence of odd numbers \(n \equiv \pm 2\;(\bmod \;5)\), which are both Fermat and Lucas pseudoprimes (in short, FL‑pseudoprimes). A Fermat pseudoprime to base a is composite number n satisfying the condition \({{a}^{{n - 1}}} \equiv 1\)(mod n). Base a is chosen to be equal to 2. A Lucas pseudoprime is a composite n satisfying \({{F}_{{n - e(n)}}} \equiv 0\)(mod n), where e(n) is the Legendre symbol \(e(n) = \left( \begin{gathered} n \hfill \\ 5 \hfill \\ \end{gathered} \right)\) and \({{F}_{m}}\) is the mth term of the Fibonacci series. According to the Baillie PSW conjecture, there are no FL pseudoprimes. If the conjecture is true, the combined primality test checking Fermat and Lucas conditions for odd numbers not divisible by 5 gives the correct answer for all numbers of the form \(n \equiv \pm 2\;(\bmod \;5)\), which generates a new deterministic polynomial primality test detecting the primality of 60 percent of all odd numbers in just two checks. In this work, we continue the study of FL pseudoprimes, started in our article “On a combined primality test” published in Russian Mathematics in 2012. We have established new restrictions on probable FL pseudoprimes and described new algorithms for checking FL primality, and, using them, we proved the absence of such numbers up to the boundary \(B = {{10}^{{21}}}\), which is more than 30 times larger than the previously known boundary 2⁶⁴ found by Gilchrist in 2013. An inaccuracy in the formulation of Theorem 4 in the mentioned article has also been corrected.