Integer properties of a composition of exponential generating functions

TL;DR

This paper explores the properties of a composition of exponential generating functions, revealing new insights that help distinguish prime numbers from composite numbers and deriving known and new properties of Bell and Euler numbers.

Contribution

It introduces new properties of composed exponential generating functions that aid in primality testing and extends understanding of Bell and Euler numbers.

Findings

New properties of exponential generating function compositions.

Characterization of prime numbers using these properties.

Derived new properties of Euler numbers.

Abstract

In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the result of paper we get the known properties of the Bell numbers(Touchard's Congruence for $k=0$) and new properties of the Euler numbers. Key words: exponential generating function, composition of generating functions, composita, primality, Touchard's Congruence, Bell numbers, Euler numbers.