The object of the paper are the so-called “unimaginable numbers”. In particular, we deal with some arithmetic and computational aspects of the Knuth’s powers notation and move some first steps into the investigation of their density. Many authors adopt the convention that unimaginable numbers start immediately after 1 googol which is equal to, and G.R. Blakley and I. Borosh have calculated that there are exactly 58 integers between 1 and 1 googol having a nontrivial “kratic representation”, i.e., are expressible nontrivially as Knuth’s powers. In this paper we extend their computations obtaining, for example, that there are exactly 2 893 numbers smaller than with a nontrivial kratic representation, and we, moreover, investigate the behavior of some functions, called krata, obtained by fixing at most two arguments in the Knuth’s power.

On the arithmetic of Knuth’s powers and some computational results about their density / Caldarola, F.; D'Atri, G.; Mercuri, P.; Talamanca, V.. - 11973:(2020), pp. 381-388. (Intervento presentato al convegno 3rd Triennial International Conference and Summer School on Numerical Computations: Theory and Algorithms, NUMTA 2019 tenutosi a ita) [10.1007/978-3-030-39081-5_33].

On the arithmetic of Knuth’s powers and some computational results about their density

d'Atri G.;Mercuri P.;Talamanca V.
2020

Abstract

The object of the paper are the so-called “unimaginable numbers”. In particular, we deal with some arithmetic and computational aspects of the Knuth’s powers notation and move some first steps into the investigation of their density. Many authors adopt the convention that unimaginable numbers start immediately after 1 googol which is equal to, and G.R. Blakley and I. Borosh have calculated that there are exactly 58 integers between 1 and 1 googol having a nontrivial “kratic representation”, i.e., are expressible nontrivially as Knuth’s powers. In this paper we extend their computations obtaining, for example, that there are exactly 2 893 numbers smaller than with a nontrivial kratic representation, and we, moreover, investigate the behavior of some functions, called krata, obtained by fixing at most two arguments in the Knuth’s power.
2020
3rd Triennial International Conference and Summer School on Numerical Computations: Theory and Algorithms, NUMTA 2019
Algebraic recurrences; Computational number theory; Knuth up-arrow notation; Unimaginable numbers
04 Pubblicazione in atti di convegno::04b Atto di convegno in volume
On the arithmetic of Knuth’s powers and some computational results about their density / Caldarola, F.; D'Atri, G.; Mercuri, P.; Talamanca, V.. - 11973:(2020), pp. 381-388. (Intervento presentato al convegno 3rd Triennial International Conference and Summer School on Numerical Computations: Theory and Algorithms, NUMTA 2019 tenutosi a ita) [10.1007/978-3-030-39081-5_33].
File allegati a questo prodotto
File Dimensione Formato  
Caldarola_Arithmetic_2020.pdf

Open Access dal 01/02/2023

Note: link all'articolo: https://www.springerprofessional.de/en/on-the-arithmetic-of-knuth-s-powers-and-some-computational-resul/17699700
Tipologia: Documento in Post-print (versione successiva alla peer review e accettata per la pubblicazione)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 260.48 kB
Formato Adobe PDF
260.48 kB Adobe PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1598637
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 8
  • ???jsp.display-item.citation.isi??? 8
social impact