Salem numbers less than 49/37

TL;DR

This paper presents a new list of Salem numbers less than 49/37 using integer linear programming and root sampling, aiming to improve understanding and verify the completeness of existing lists.

Contribution

It introduces a novel method combining integer linear programming and root sampling to identify Salem numbers below a specific bound, expanding current knowledge.

Findings

New Salem numbers identified below 49/37

Potential validation of existing Salem number lists

Method may reveal completeness of current classifications

Abstract

A certain number of lists of small Salem numbers, some of which are certified as complete, are available online. Notably, the website of M. J. Mossinghoff features a list of 47 Salem numbers smaller than 1.3, as well as complete lists of Salem numbers of fixed degrees that are smaller than various bounds. The objective of this work is to advance the understanding of Salem numbers by providing a list of Salem numbers smaller than a threshold of 49/37 (≈ 1.324324) through the implementation of a method based on integer linear programming and uniform sampling of root separators. Beyond the intrinsic interest of the newly detected Salem numbers, the rediscovery of already known Salem numbers through this alternative method could offer valuable insights into the potential completeness of already existing lists for degrees where it has not yet been proven.