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On Machine Integers and Arithmetic

Authors:
Pavel Loskot

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

Abstract:
All signal and data processing is performed on computing machines. However, the computing efficiency requires that numbers are represented in a finite memory space. It is claimed that all such numbers can be considered to be integers, and that decimal point has purely syntactical meaning to align numbers in arithmetic operations. This subtle, but fundamental observation seems to have been ignored so far. As an introductory exploration of integer arithmetic, this paper introduces a dual modulo operator to select digits in string representations of machine numbers. Moreover, it is proposed that natural integers offset by a real-valued constant satisfy Peano axioms. The Fermat last theorem is then considered as an example of Diophantine equation. It is shown how it can be modified to allow the solutions to exist. A Fermat metric is newly introduced to define distances between integers to allow their partitioning into subsets. These results point at the importance of investigating integer arithmetic, integer algebra, and integer analysis in designing and modeling computing systems.

Pages: 63 to 66

Copyright: Copyright (c) IARIA, 2023

Publication date: March 13, 2023

Published in: conference

ISSN: 2519-8432

ISBN: 978-1-68558-057-5

Location: Barcelona, Spain

Dates: from March 13, 2023 to March 17, 2023