Representations for complex numbers with integer digits

Perfect Numbers with Identical Digits

Suppose g ≥ 2. A natural number N is called a repdigit in base g if it has the shape a g −1 g−1 for some 1 ≤ a < g, i.e., if all of its digits in its base g expansion are equal. The number N is called perfect if σ(N) = 2N , where σ(N) := � d|N d is the usual sum of divisors function. We show that in each base g, there are at most finitely many repdigit perfect numbers, and the set of all such n...

Developing a Mixed Integer Quadratic Programing Model with Integer Numbers for Designing a Dynamic closed-loop Logistics Network

Logistics Network Design includes network configuration decisions having long-standing influences on other tactical and operational decisions. Recently, regarding environmental issues and customer awareness and global warming closed-loop supply chain network design is taken into consideration. The proposed network for the integrated forward and reverse logistics is developed by formulating a cy...

On Binary Representations of Integers with Digits

Güntzer and Paul introduced a number system with base 2 and digits −1, 0, 1 which is characterized by separating nonzero digits by at least one zero. We find an explicit formula that produces the digits of the expansion of an integer n which leads us to many generalized situations. Syntactical properties of such representations are also discussed. 1. A binary number system Integers n can be wri...

Β-expansions with Deleted Digits for Pisot Numbers Β

An algorithm is given for computing the Hausdorff dimension of the set(s) Λ = Λ(β,D) of real numbers with representations x = ∑∞ n=1 dnβ −n, where each dn ∈ D, a finite set of “digits”, and β > 0 is a Pisot number. The Hausdorff dimension is shown to be log λ/ log β, where λ is the top eigenvalue of a finite 0-1 matrix A, and a simple algorithm for generating A from the data β,D is given.

beta-EXPANSIONS WITH DELETED DIGITS FOR PISOT NUMBERS beta