Here, we show that there is no integer $s\ge 3$ such that the sum of $s$th powers of two consecutive Fibonacci numbers is a Fibonacci number.

For a fixed prime $l\in \mathbf{Z}$, we consider zeta functions for certain types of (not necessarily commutative) algebras over the completion $\mathbf{Q}_{l}$ of $\mathbf{Q}$ and show that they satisfy several properties analogous to those of the usual Hasse-Weil zeta function of an algebraic variety over a finite field.

Recently Yousefi and Hesameddini [13] have obtained a characterization for shift invariant subspaces of a special class of Hilbert spaces contained in the Hardy space $H^2$. In the present note we settle an open problem posed by them in their paper. In fact, by discussing invariance under multiplication by finite Blaschke factors we prove a far more general result than the main result of [13]. We prove our results under much weaker assumptions than the assumptions of [13] and with a simpler proof.