Partitions into powers of an algebraic number. (English) Zbl 07846623
In this article, the authors discuss the investigation of general partitions as an analogous, further they study partitions of complex number as sum of positive powers of 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 partition function attains all positive integer as values when \(\beta\) satisfyies certain conditions. The necessary proofs are given in the form of two theorems and some related results. The authors also proposed four open problems, but one problem was solved by 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.
Reviewer: M. P. Chaudhary (New Delhi)
MSC:
| 11P81 | Elementary theory of partitions |
| 11P84 | Partition identities; identities of Rogers-Ramanujan type |
| 11R11 | Quadratic extensions |
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