On some properties of Fermat numbers (Q1406433)

Let \(\,F_m = 2^{2^m}+1\), \(\,m=0,1,2,\dots\,\) be the Fermat numbers. According to \textit{A. K. Lenstra, H. W. Lenstra, M. S. Manasse} and \textit{J. M. Pollard} [Math. Comput. 61, 319--349 (1993; Zbl 0792.11055)] the first five Fermat numbers are prime and \(F_5\) and all next \(F_m\) which were tested proved to be composite. The author sets out some tests to check whether the Fermat numbers are prime.