On simultaneous number systems with \(3\) bases

Abstract. Simultaneous number systems with 2 bases were studied before in different structures. One way of generalization is using 3 bases instead of 2. In this paper we consider the ring of integers of the quadratic field \(\mathbb{Q}[\sqrt{5}]\). The difficulty is the amount of triplets we have to check to validate the number system property, so the development of a software which can check these sets in acceptable times (using parallelization, GPU computing) was necessary. The existence of simultaneous number systems with three bases was confirmed by our software for some concrete choices of bases and digit sets.