Canonical number systems can be viewed as natural generalizations of radix representations of ordinary integers to algebraic integers. A slightly modified version of an algorithm of B. Kovács and A. Pethő is presented here for the determination of canonical number systems in orders of algebraic number fields. Using this algorithm canonical number systems of some quartic fields are computed.

It is already known that all Pisot numbers are beta numbers, but for Salem numbers this was proved just for the degree 4 case. In 1945, R. Salem showed that for any Pisot number θ we can construct a sequence of Salem numbers which converge to θ. In this short note, we give some results on the beta expansion for infinitely many sequences of Salem numbers obtained by this construction.

In a recent paper we proved that there are at most finitely many complex numbers $t\ne 0,1$ such that the points $(2,\sqrt{2(2-t)})$ and $(3,\sqrt{6(3-t)})$ are both torsion on the Legendre elliptic curve defined by $y^2=x(x-1)(x-t)$. In a sequel we gave a generalization to any two points with coordinates algebraic over the field $\mathbf{Q}\left(t\right)$ and even over $\mathbf{C}\left(t\right)$. Here we reconsider the special case $(u,\sqrt{u(u-1)(u-t)})$ and $(v,\sqrt{v(v-1)(v-t)})$ with complex numbers $u$ and $v$.

The aim of this paper is to present some computer data suggesting the correct size of bounds for exponential sums of Fourier coefficients of holomorphic cusp forms.

A bibliography of recent papers on lower bounds for discriminants of number fields and related topics is presented. Some of the main methods, results, and open problems are discussed.