Nombres de Pisot, nombres de Salem et la conjecture de Lehmer

Abstract

We investigate the relationship between the set S of Pisot numbers and the set T of Salem numbers. Salem first established that: " every Pisot number is an accumulation point of the set T ". Building on Boyd's method, we show that every accumulation point of T belongs to S. Together, these results imply that the union S U T forms a closed subset of the real half-line ]1,+infinity[. Consequently, this settles Boyd's conjecture while disproving Lehmer's conjecture.