Radix Representation Research Articles

Conversion of modular numbers to their mixed radix representation by a matrix formula

Introduction. Let mi > 1, (i = 1, 2, * , s), be integers relatively prime in pairs and denotem = m1M2 m. *Ms . If xi,O < xi < mi, (i = 1, 2, * , s) are integers, the ordered set (x1 , x2 , x8) is called a modular number, with respect to the moduli mi (i = 1, 2, . , s) and it denotes a unique residue class mod m. Modular arithmetic has been developed [1], [2], [5], and its use in computers has been suggested [1], [5]. It has also been applied in the solution of various problems [2], [6]. A central question is to determine the least nonnegative residue mod m of a given residue class (xl , x2, * *, x8). Denote it by n. In order to work entirely in the given modular system it was suggested [1], [3], [7] and [8] to obtain n in its mixed radix representatioin with respect precisely to the radices mi (i = 1, 2, , s), thus in the form

On modular computation

It will be assumed that the reader is familiar w,ith the elementary theory of congruences [1]. Let in, M2, ... *, mn be s integers relatively prime in pairs and let Ml =MIM2 im8. Let xl, x2, * * , x8be an ordered set of s integers such that 0 < xi < mi. There exists one and only one residue class x mod 31 such that x xi (mod mi) andwethereforewritex=(xl,x2, , X8).Ifx= (xl, ,x8),y= (y1,* ,Y.) then x y = (x1 i Yy, * * , x8 4 y), xy = (x1y1, , x.y8), where the ith coordinates must be reduced mod mi. The symbols (xi, * , x,) are called modular numbers, but it should be kept in mind that (xi, * , x,) does not denote a number but a residue class mod Ml. The purpose of this note is to describe a simple iterative procedure to determine the least non-negative residue mod Mll of a given residue class (xl, * * *, x,). The iteration process described below gives the least non-negative residue in mixed radix representation. Notation. The moduli are denoted by mi, m *, in8. We put mo = 1,