Periodicity of the P-adic Expansion after Arithmetic Operations in P-adic Field

Abstract

For the past few years, we have been developing the Exact Scientific Computational Library (ESCL) using P-adic arithmetic. The effort has been focusing on converting all rational number operations to integer calculation, and fully taking the advantage of fast integer multiplication of modern computer architectures. By properly selecting prime numbers as the bases and practically choosing the length r for P-adic expansion of rational numbers, we have shown some promising results for large matrix operations. One problem that we have confronted is the overflow of the prescribed bound in P-adic arithmetic, Hensel code originally defined by Krishnamurthy, Rao, and Subramanian [3] may give an invalid Hensel code after arithmetic operations if the length r of the P-adic expansion is not properly chosen. In this paper, we will show and prove the periodicity after arithmetic operations in P-adic number systems. If we can represent all the P-adic sequences with a complete period during all the calculations, then we are sure to be able to carry out all the arithmetic operations exactly. Two formulas for the maximum length of the P-adic expansion's periodic part after the P-adic arithmetic operations are given in this paper.