In this note we consider projective limits of Sinnott and Washington groups of circular units in the cyclotomic ${\mathbb{Z}}_p$-extension of an abelian field. A concrete example is given to show that these two limits do not coincide in general.

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S(a,p^{\alpha }+1):={\sum }_{x{\in }_q}\chi \left(a{x}^{p^{\alpha }+1}\right)$ where χ is a non-trivial additive character of the finite field ${}_q$, $q=p^e$ odd, and $a\in {*}_q$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^{\alpha }+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain...

We show that any factorization of any composite Fermat number $F_m={2^2}^m+1$ into two nontrivial factors can be expressed in the form $F_m=(k{2}^n+1)(\ell 2^n+1)$ for some odd $k$ and $\ell ,k\ge 3,\ell \ge 3$, and integer $n\ge m+2,3n<2^m$. We prove that the greatest common divisor of $k$ and $\ell$ is 1, $k+\ell \equiv 0mod{2}^n,max(k,\ell )\ge F_{m-2}$, and either $3|k$ or $3|\ell$, i.e., $3h{2}^{m+2}+1|F_m$ for an integer $h\ge 1$. Factorizations of $F_m$ into more than two factors are investigated as well. In particular, we prove that if $F_m={(k{2}^n+1)}^2(\ell 2^j+1)$ then $j=n+1,3|\ell$ and $5|\ell$.