Home // International Journal On Advances in Systems and Measurements, volume 16, numbers 3 and 4, 2023 // View article

Finite Memory Arithmetic and the Number Representations on Computing Machines

Authors:
Pavel Loskot

Keywords: dual modulo arithmetic; Fermat last theorem; Fermat metric; natural numbers

Abstract:
Numerical values on computing machines are normally represented as finite-memory data objects. Such values can be mapped to a finite set of integers, so the computing machines effectively perform only the integer arithmetic operations. Consequently, all programs and algorithms implemented on the computing machines can be modeled by Diophantine equations. This is also true for the computations that are performed by analog computers with a limited resolution. In order to study the Diophantine systems, this paper introduces a dual modulo operator to select the subsets of digits in the string representations of machine numbers, which is also useful when the number equality is replaced by a modulo equivalence. Moreover, it is shown that the solutions of Diophantine equations such as the Fermat Last Theorem can be obtained in the domain of integers that are offset by the same constant real value. The Fermat metric is newly introduced to define the distances between integers and other discrete sets of numbers. Finally, a two-dimensional quantization is devised for mixed arithmetic operations to allow the computations to be equivalently performed either between discrete analog values, or between the integer indices. The key claim of this paper is that all practical computing problems can be exactly and completely represented by an integer arithmetic.

Pages: 150 to 158

Copyright: Copyright (c) to authors, 2023. Used with permission.

Publication date: December 30, 2023

Published in: journal

ISSN: 1942-261x