P-adic number theory

Abstract

p-adic systems, although introduced by Hensel in 1908, have only recently attracted attention for their possible uses in exact linear computations, matrix processors, signal transformations and cryptography. The paper gives a detailed analysis of the subject of p-adic number theory. An examination of both the infinite and finite p-adic number systems is presented. For infinite systems, a practical efficient algorithm is developed for the computation of the p-adic period for any rational number, given the prime p. In finite systems, on the other hand, we demonstrate that the original algorithms developed by Krishnamurthy for the four main arithmetic operations are erroneous and we present new algorithms which circumvent these drawbacks.