Advertisement

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
  • 23 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11973)

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

Keywords

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

Notes

Aknowledgments

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.

References

  1. 1.
    Ackermann, W.: Zum hilbertschen aufbau der reellen zahlen. Math. Ann. 99, 118–133 (1928).  https://doi.org/10.1007/BF01459088MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bennett, A.A.: Note on an operation of the third grade. Ann. Math. Second Series 17(2), 74–75 (1915).  https://doi.org/10.2307/2007124MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blakley, G.R., Borosh, I.: Knuth’s iterated powers. Adv. Math. 34(2), 109–136 (1979).  https://doi.org/10.1016/0001-8708(79)90052-5MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blakley, G.R., Borosh, I.: Modular arithmetic of iterated powers. Comp. Math. Appl. 9(4), 567–581 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bowers, J.: Exploding Array Function. Accessed 25 Apr 2019. http://www.polytope.net/hedrondude/array.htm
  6. 6.
    Bowers, J.: Extended operator notation. Accessed 21 Apr 2019. https://sites.google.com/site/largenumbers/home/4-1/extended_operators
  7. 7.
    Bromer, N.: Superexponentiation. Mathematics Magazine 60(3), 169–174 (1987). JSTOR 2689566MathSciNetCrossRefGoogle Scholar
  8. 8.
    Caldarola, F.: The exact measures of the Sierpiński \(d\)-dimensional tetrahedron in connection with a Diophantine nonlinear system. Commun. Nonlinear Sci. Numer. Simul. 63, 228–238 (2018). https://doi.org/10.10.16/j.cnsns.2018.02.026
  9. 9.
    Caldarola, F., Cortese, D., d’Atri, G., Maiolo, M.: Paradoxes of the infinite and ontological dilemmas between ancient philosophy and modern mathematical solutions. In: Sergeyev, Y.D., Kvasov, D.E. (eds.) NUMTA 2019. LNCS 11973, pp. 358–372. Springer, Cham (2020).  https://doi.org/10.1007/978-3-030-39081-5_31
  10. 10.
    Caldarola, F., d’Atri, G., Maiolo, M.: What are the "unimaginable numbers"? (submitted)Google Scholar
  11. 11.
    Donner, J., Tarski, A.: An extended arithmetic of ordinal numbers. Fundamenta Mathematicae 65, 95–127 (1969)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Friedman, H.M.: Long finite sequences. J. Comb. Theor. Series A 95(1), 102–144 (2001).  https://doi.org/10.1006/jcta.2000.3154MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goodstein, R.L.: Transfinite ordinals in recursive number theory. J. Symbolic Logic 12(4), 123–129 (1947).  https://doi.org/10.2307/2266486MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hooshmand, M.H.: Ultra power and ultra exponential functions. Integr. Transforms Spec. Functions 17(8), 549–558 (2006).  https://doi.org/10.1080/1065246050042224MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Knobel, R.A.: Exponentials reiterated. American Mathematical Monthly 88(4), 235–252 (1981).  https://doi.org/10.2307/2320546MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Knuth, D.E.: Mathematics and computer science: coping with finiteness. Science 194(4271), 1235–1242 (1976).  https://doi.org/10.1126/science.194.4271.1235MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lameiras Campagnola, M., Moore, C., Costa, J.F.: Transfinite ordinals in recursive number theory. J. Complex. 18(4), 977–1000 (2002).  https://doi.org/10.1006/jcom.2002.0655CrossRefGoogle Scholar
  18. 18.
    Leonardis, A., d’Atri, G., Caldarola, F.: Beyond Knuth’s notation for "Unimaginable Numbers" within computational number theory. arXiv:1901.05372v2 [cs.LO] (2019)
  19. 19.
    Littlewood, J.E.: Large numbers. Math. Gaz. 32(300), 163–171 (1948).  https://doi.org/10.2307/3609933CrossRefGoogle Scholar
  20. 20.
    MacDonnell, J.F.: Some critical points of the hyperpower function \({x^{x^{\dots }}} x^{x^{\dots }}\). Int. J. Math. Educ. 20(2), 297–305 (1989).  https://doi.org/10.1080/0020739890200210MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marshall, A.J., Tan, Y.: A rational number of the form \(^aa\) with \(a\) irrational. Math. Gaz. 96, 106–109 (2012)CrossRefGoogle Scholar
  22. 22.
    Munafo, R.: Large Numbers at MROB. Accessed 19 May 2019Google Scholar
  23. 23.
    Nambiar, K.K.: Ackermann functions and transfinite ordinals. Appl. Math. Lett. 8(6), 51–53 (1995).  https://doi.org/10.1016/0893-9659(95)00084-4MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

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

Personalised recommendations