Beyond Knuth's notation for unimaginable numbers within computational number theory

Year 2022, Volume: 31 Issue: 31, 55 - 73, 17.01.2022

Antonino Leonardıs Gianfranco D'atrı Fabio Caldarola

https://doi.org/10.24330/ieja.1058413

Cited By: 5

Abstract

Literature considers under the name "unimaginable numbers" any
positive integer going beyond any physical application. One of the most known
methodologies to conceive such numbers is using hyper-operations, that is a
sequence of binary functions dened recursively starting from the usual chain:
addition - multiplication - exponentiation. The most important notations to
represent such hyper-operations have been considered by Knuth, Goodstein,
Ackermann and Conway as described in this work's introduction. Within this
work we will give an axiomatic setup for this topic, and then try to nd on one
hand other ways to represent unimaginable numbers, as well as on the other
hand applications to computer science, where the algorithmic nature of representations and the increased computation capabilities of computers give the
perfect eld to develop further the topic, exploring some possibilities to effectively operate with such big numbers. In particular, we will give some axioms
and generalizations for the up-arrow notation and, considering a representation via rooted trees of the hereditary base-n notation, we will determine in
some cases an effective bound related to "Goodstein sequences" using Knuths
notation. Finally, we will also analyze some methods to compare big numbers,
proving specically a theorem about approximation using scientic notation
and a theorem on hyperoperation bounds for Steinhaus-Moser notation.

Keywords

Computational number theory, unimaginable numbers, Knuth, big data, number representation, Goodstein

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