On the Arithmetic of Knuth’s Powers and Some Computational Results About Their Density

  • Fabio Caldarola
  • Gianfranco d’Atri
  • Pietro Mercuri
  • Valerio TalamancaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11973)


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 \(10^{100}\), 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 \(10^{10\,000}\) 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 \(a\!\uparrow ^b\!c\).


Unimaginable numbers Knuth up-arrow notation Algebraic recurrences Computational number theory 



This work is partially supported by the research projects “IoT&B, Internet of Things and Blockchain”, CUP J48C17000230006, POR Calabria FESR-FSE 2014–2020. The third author was also supported by the research grant “Ing. Giorgio Schirillo” of the Istituto Nazionale di Alta Matematica “F. Severi”, Rome.


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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversità della CalabriaArcavacata di RendeItaly
  2. 2.Università “Tor Vergata”RomeItaly
  3. 3.Università Roma TreRomeItaly

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