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A real-world application of branching processes to a problem in cell biology where the generalized Fibonacci numbers known as knacci numbers play a crucial role is described. The k-nacci sequence is used to obtain asymptotics, computational formulas, and to justify certain practical simplifications. Along the way, an explicit formula for the sum of k-nacci numbers is established.

By computing the rank of the group of unimodular units in a given number field, we provide a simple proof of the classification of the number fields containing algebraic integers of modulus 1 that are not roots of unity. For a number field K, let V K denote the set of algebraic integers in K of modulus 1. Such numbers are necessarily units in K: if u ∈ K is integral and |u| = 1, then u = u −1 is also an integral element of K. Therefore V K is a subgroup of the unit group U K of K. Since U K is a finitely generated abelian group, so too is V K . According to Dirichlet's unit theorem, the rank of U K is determined in a simple way by the signature of K, and one is led to wonder whether there is an equally simple way to determine the rank of V K . In this note we show that this is indeed the case.A natural question to ask is when V K properly contains the group W K of roots of unity in K. That is, when does K contain algebraic integers of modulus 1 that are not roots of unity? In 1975, MacCluer and Parry [2] partially answered this question by proving that if K is a Galois extension of Q then W K = V K if and only if K is imaginary and not a CMfield (defined below). That same year Parry [3] extended this result, with slightly more complicated hypotheses, to all number fields. It turns out that both of these results are

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