Partitions into powers of an algebraic number (Q6536864)

In this article, the authors discuss general partitions, further they study partitions of complex number as the sum of positive powers of a fixed algebraic number \(\beta\), and they show that when \(\beta\) is real quadratic then the number of partitions are always finite if and only if some conjugate of \(\beta > 1\). Also they discuss when the partition function attains all positive integer values when \(\beta\) satisfies certain conditions. The proofs are given in the form of two theorems and some related results. The authors also propose four open problems, but one problem was solved by \textit{A. Dubickas} [``Representations of a number in an arbitrary base with unbounded digits'', Georgian Math. J. (to appear)], when the this article was under review.