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

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 defined 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 find 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 field 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 Knuth's notation. Finally, we will also analyze some methods to compare big numbers, proving specifically a theorem about approximation using scientific notation and a theorem on hyperoperation bounds for Steinhaus-Moser notation.

Some Variants of Integer Multiplication

“…Using the same argument, the multiples on the right of 1 when t → 0 − must coincide with the usual ones represented in (6). But now, if t → 0 − , then c 2 → +∞.…”

“…We consider the usual representation of the set of integers Z in the number line as the set {1 0 x : x ∈ Z}. We can see this representation in (6). Now, if we see t as a real variable, we can study the set {1 t x : x ∈ Z} when t tends to 0.…”

“…For instance, if we want to write 93 as the sum of 6 terms of an arithmetic progressionwhose difference is 5, then the quotient indicates to us the first term of the solution: 93 5 6 = 93/6 − 5 • 5/2 = 3. Then, 93 = 3 5 6 = 3 + 8 + 13 + 18 + 23 + 28.…”

“…This manuscript deals with typical themes of the Pythagorean mathematical heritage. We recommend to the interested reader in this topic to investigate the bibliography contained in [5,6].…”

A Note on the Geometry of Closed Loops

“…The Ramsey theory/approach shows potential for a diversity of mathematical and physical applications, including graph theory [7], ergodic theory [7][8][9], unimaginable numbers, Goodstein sequences, Knuth powers, Planar geometry [16], labeled graphs [17], theory of dynamic billiards [12][13][14]18], statistical physics [19], axiomatic thermodynamics [20] and analysis of many-body interactions [21]. We demonstrate that the Ramsey theory enables the instructive analysis of a discrete sets of points located on closed curves.…”