Weber's Class Number Problem in the Cyclotomic $\Z_2$-Extension of $\Q$

Abstract

Let $h_n$ denote the class number of $\Q(2\cos(2\pi/2^{n+2}))$. Weber proved that $h_n$ is odd for all $n\geq 1$. We claim that if $\ell$ is a prime number less than $10^7$, then for all $n\geq 1$, $\ell$ does not divide $h_n$.