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H.J. Zassenhaus conjectured that any unit of finite order in the integralgroup ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rationalgroup algebra $\mathbb{Q}G$ to an element of the form $\pm g$ with $g \in G$.Though known for some series of solvable groups, the conjecture has been provedonly for thirteen non-abelian simple groups. We prove the Zassenhaus Conjecturefor the groups $\operatorname{PSL}(2,p)$, where $p$ is a Fermat or Mersenneprime. This increases the list of non-abelian simple groups for which theconjecture is known by probably infinitely many, but at least by 49, groups.Our result is an easy consequence of known results and our main theorem whichstates that the Zassenhaus Conjecture holds for a unit in$\mathbb{Z}\operatorname{PSL}(2,q)$ of order coprime with $2q$, for some primepower $q$.

Zassenhaus Conjecture on torsion units holds for $\operatorname{PSL}(2,p)$ with $p$ a Fermat or Mersenne prime