New properties for a composition of some generating functions for primes

TL;DR

This paper explores new properties of composed generating functions with logarithmic and ordinary functions, leading to novel primality criteria for special number sequences like Mersenne and Lucas numbers.

Contribution

It introduces new properties of generating function compositions that can be used to develop primality tests for various special numbers.

Findings

Derived primality criteria for Mersenne numbers

Established properties for compositions involving Lucas and Pell-Lucas numbers

Proposed methods to distinguish prime from composite numbers using generating functions

Abstract

In this paper, we consider properties of coefficients of a generating functions composition, where the outer function is a logarithmic generating function and the inner function is an ordinary generating function with integer coefficients. Using notions of composita and composition of generating functions, we get new properties for this composition. The properties can be used for distinguishing prime numbers from composite numbers. As an application, obtained results can be used to obtain new primality criteria. We obtain primality criteria for the Mersenne numbers, the Lucas numbers, the Pell-Lucas numbers, the Jacobsthal-Lucas numbers, and the Lucas sequences. Keywords: generating function, composition of generating function, composita, primality criterion.