Carry propagation in signed digit representations

Authors:
Clemens Heuberger;Helmut Prodinger
Affiliations:
Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria;The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, P.O. Wits, 2050 Johannesburg, South Africa
Venue:
European Journal of Combinatorics
Year:
2003

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Abstract

Von Neumann's addition method adds two numbers given in q-ary representation by forming a number consisting of the added digits, reduced modulo q, and another number, representing the carries and repeating this until the string of carries consists only of zeros. The average number of iterations was studied by Knuth.We extend these results by considering the (q,d) system, with base q and digits d, d + 1, ..., d + q - 1, as well as the symmetric signed digit expansions, for even q, with digits -q/2,...,q/2, and a special rule to make representations of integers unique.