University of IsfahanInternational Journal of Group Theory2251-765013320240901Orders of simple groups and the Bateman--Horn Conjecture2572692752110.22108/ijgt.2023.136666.1828ENGareth AneurinJonesDepartment of Mathematics, School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UKAlexander K.ZvonkinLaBRI, UniversiteĢ de Bordeaux, 351 Cours de la LibeĢration, F-33405, Talence, FranceJournal Article20230208We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and H\"older in the 1890s.) The groups satisfying this condition are ${\rm PSL}_2(8)$, ${\rm PSL}_2(9)$ and ${\rm PSL}_2(p)$ for primes $p$ such that $p^2-1$ is a product of six primes. The conjecture suggests that there are infinitely many such primes $p$, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, the construction of highly symmetric block designs, and the possible existence of infinitely many K$n$ groups for each $n\ge 5$.https://ijgt.ui.ac.ir/article_27521_b123f0c513f61f9c023d31a0e1bfa7c8.pdf