Canonical Diophantine representations of natural numbers with respect to quadratic “bases”

In 1957, Bergman proved that every natural number can be expressed uniquely as a sum of distinct, non-consecutive integral powers of . More recently, in 2009, Gerdemann showed how such a decomposition of n leads to the corresponding “Zeckendorf representation” of nFm, for all sufficiently large m, in which Fm denotes the mth Fibonacci number. Here we extend these results by replacing φ with an arbitrary quadratic irrational real number α. In this general setting, we will connect our new “base α” representation of n to Ostrowskiʼs decomposition of nqm, in which qm denotes the mth continuant associated with α.