# The necessary and sufficient condition for an algebraic integer to be a Salem number

## TL;DR

This paper establishes a precise criterion for identifying Salem numbers among roots of certain reciprocal polynomials and analyzes the probability of this condition holding for powers of these roots.

## Contribution

It provides a necessary and sufficient condition for roots to be Salem numbers and evaluates the likelihood of this condition for powers of the roots.

## Key findings

- Derived a complete criterion for Salem numbers from reciprocal polynomials
- Calculated the probability that the condition holds for powers of the roots
- Enhanced understanding of Salem number distribution and properties

## Abstract

We present a necessary and sufficient condition for a root greater than unity of a monic reciprocal polynomial of an even degree at least four, with integer coefficients, to be a Salem number. We determine the probability of fulfillment the condition for an arbitrary power of the root.