University of Primorska
Faculty of Mathematics, Natural Sciences and Information Technologies
The article by Assoc. Prof. Dr. Slobodan Filipovski from the Department of Mathematics, entitled “On the Asymptotic Density of Fermat Pseudoprimes to a Fixed Base,” has been published in The American Mathematical Monthly.
Prime numbers are fundamental objects in number theory, and identifying them also plays an important role in modern cryptography. One of the classical approaches to testing primality is based on Fermat’s Little Theorem. However, some composite numbers pass this test; these are known as Fermat pseudoprimes.
In his article On the Asymptotic Density of Fermat Pseudoprimes to a Fixed Base, Filipovski revisits a classical result stating that the set of Fermat pseudoprimes to a fixed base has asymptotic density zero. This result was originally proved by Paul Erdős in 1956, one of the most influential mathematicians of the 20th century, using a substantially more sophisticated argument. Filipovski demonstrates that the same result can be proved considerably more concisely and using more elementary methods. The key idea behind the new approach is to reduce the problem to a significantly simpler counting argument through an appropriate partitioning of numbers and a few elementary estimates. This makes it possible to establish Erdős’s classical result on the asymptotic sparsity of Fermat pseudoprimes with a much shorter proof and without relying on more advanced technical machinery.
The article was published in September 2026 in The American Mathematical Monthly, one of the most renowned mathematical journals, with a tradition spanning more than a century. Throughout its history, the journal has featured contributions from numerous leading mathematicians, with a particular emphasis on elegant ideas and results of broader mathematical significance. The journal is highly selective, accepting approximately one-tenth of submitted manuscripts. Publication in the journal therefore represents significant international recognition of the quality and clarity of the mathematical work. Taylor & Francis Online .
This is not Filipovski’s first contribution to the journal. In 2022, his Problem 12354, concerning the zeros of the polynomial \(x^n+x^{n-1}+\cdots+x-k\), was also published in the Problems and Solutions section.
Read the article here: On the Asymptotic Density of Fermat Pseudoprimes to a Fixed Base
