Polygonal radix representations of complex numbers

Polygonal Numbers

In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = ∆ + ∆ + ∆”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar lang...

Redundant Radix Representations of Rings

ÐThis paper presents an analysis of radix representations of elements from general rings; in particular, we study the questions of redundancy and completeness in such representations. Mappings into radix representations, as well as conversions between such, are discussed, in particular where the target system is redundant. Results are shown valid for normed rings containing only a finite number...

Radix Representations of Quadratic Fields

Let p be an algebraic integer in a quadratic number field whose minimum polynomial is x1 + p, x + p,,. Then all the elements of the ring Z [ p] can be written uniquely in the base p as CFS:=o a,pk, where 0 < ak < 1 pa 1, if and only ifp, > 2 and-1 <P,<Po. 1. 1NTR00uc-1~0~ Various bases have been proposed for writing complex numbers in positional notation as a single string of digits, without se...

On Universal Sums of Polygonal Numbers

For m = 3, 4, . . . , the polygonal numbers of order m are given by pm(n) = (m−2) ` n 2 ́ +n (n = 0, 1, 2, . . . ). For positive integers a, b, c and i, j, k > 3 with max{i, j, k} > 5, we call the triple (api, bpj , cpk) universal if for any n = 0, 1, 2, . . . there are nonnegative integers x, y, z such that n = api(x)+bpj(y)+cpk(z). We show that there are only 95 candidates for universal triple...

On Wiener numbers of polygonal nets