On the Baillie PSW Conjecture

IF 0.5 Q3 MATHEMATICS Russian Mathematics Pub Date : 2024-08-06 DOI:10.3103/s1066369x24700294

Sh. T. Ishmukhametov, B. G. Mubarakov, R. G. Rubtsova, E. V. Oleinikova

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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.

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关于贝利 PSW 猜想

摘要 Baillie PSW猜想提出于1980年，并以其作者Baillie, Pomerance, Selfridge, and Wagstaff, Jr.的名字命名。该猜想与存在奇数 \(n \equiv \pm 2\;(\bmod \;5)\)的问题有关，这些奇数既是费马伪素数又是卢卡斯伪素数（简言之，FL-伪素数）。以 a 为底数的费马假素数是满足条件 \({{a}^{n - 1}} \equiv 1\)(mod n) 的合成数 n。卢卡斯伪素数是满足条件（\({{F}_{n - e(n)}}} ）的合数 n。\(mod n), 其中 e(n) 是 Legendre 符号 \(n) = \left( \begin{gathered} n \hfill \5 \hfill \\end{gathered} \right)\) 并且 \({{F}_{m}}\) 是斐波纳契数列的第 m 项。根据贝利 PSW 猜想，不存在 FL 伪素数。如果猜想是真的，那么对不能被 5 整除的奇数进行费马条件和卢卡斯条件的联合原始性检验，就能对所有形式为 \(n \equiv \pm 2\;(\bmod \;5)\)的数给出正确答案，这就产生了一种新的确定性多项式原始性检验，只需两次检验就能检测出 60% 的奇数的原始性。在这项工作中，我们将继续研究 FL 伪素数，这项研究始于 2012 年发表在《俄罗斯数学》上的文章 "论组合原始性检验"。我们对可能的FL伪素数建立了新的限制，并描述了检查FL原始性的新算法，利用这些算法，我们证明了在边界\(B = {{10}^{{21}}\) 以内不存在这样的数，这比吉尔克里斯特（Gilchrist）在2013年发现的已知边界264大了30多倍。上述文章中定理 4 的表述不准确之处也得到了纠正。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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