Research Article
BibTex RIS Cite

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

Year 2022, Volume: 31 Issue: 31, 55 - 73, 17.01.2022
https://doi.org/10.24330/ieja.1058413

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.

References

  • W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann., 99(1) (1928), 118-133.
  • L. Antoniotti, F. Caldarola, G. d’Atri and M. Pellegrini, New approaches to basic calculus: an experimentation via numerical computation, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 329-342.
  • L. Antoniotti, F. Caldarola and M. Maiolo, Infinite numerical computing applied to Hilbert’s, Peano’s, and Moore’s curves, Mediterr. J. Math., 17(3) (2020), 99 (19 pp).
  • G. d’Atri, Logic-based consistency checking of XBRL instances, IJACT 3 (2014), 126-131.
  • A. A. Bennett, Note on an Operation of the Third Grade, Ann. of Math, Second Series, 17(2) (1915), 74-75.
  • J. Bowers, Exploding Array Function, Retrieved on 21-11-2018, http://www.polytope.net/hedrondude/array.htm.
  • F. Caldarola, The exact measures of the Sierpi´nski d-dimensional tetrahedron in connection with a Diophantine nonlinear system, Commun. Nonlinear Sci. and Numer. Simul., 63 (2018), 228-238.
  • F. Caldarola, The Sierpinski curve viewed by numerical computations with infinities and infinitesimals, Appl. Math. Comput., 318 (2018), 321-328.
  • F. Caldarola, D. Cortese, G. d’Atri and M. Maiolo, Paradoxes of the infinite and ontological dilemmas between ancient philosophy and modern mathematical solutions, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 358-372.
  • F. Caldarola, G. d’Atri, M. Maiolo and G. Pirillo, New algebraic and geometric constructs arising from Fibonacci numbers. In honor of Masami Ito, Soft Computing, 24(23) (2020), 17497-17508.
  • F. Caldarola, G. d’Atri, M. Maiolo and G. Pirillo, The sequence of Carboncettus octagons, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 373-380.
  • F. Caldarola, G. d’Atri, P. Mercuri and V. Talamanca, On the arithmetic of Knuth’s powers and some computational results about their density, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 381-388.
  • F. Caldarola and M. Maiolo, On the topological convergence of multi-rule sequences of sets and fractal patterns, Soft Computing, 24(23) (2020), 17737- 17749.
  • F. Caldarola, M. Maiolo and V. Solferino, A new approach to the Z-transform through infinite computation, Commun. Nonlinear Sci. Numer. Simul., 82 (2020), 105019.
  • J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, New York, 1957.
  • J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus, New York, 1996.
  • R. L. Goodstein, Transfinite ordinals in recursive number theory, J. Symbolic Logic, 12 (1947), 123-129.
  • R. L. Goodstein, On the restricted ordinal theorem, J. Symbolic Logic, 9 (1944), 33-41.
  • D. E. Knuth, Mathematics and computer Science: coping with finiteness, Science, 194(4271) (1976), 1235-1242.
  • A. Leonardis, Continued fractions in local fields and nested automorphisms, Ph.D. thesis, 2014.
  • R. Munafo, Inventing New Operators and Functions, Large Numbers at MROB, Retrieved on 19-11-2019.
  • R. Munafo, Versions of Ackermann’s Function, Large Numbers at MROB, Retrieved on 19-11-2019.
  • K. K. Nambiar, Ackermann Functions and Transfinite Ordinals, Appl. Math. Lett., 8(6) (1995), 51-53.
  • R. Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite, Princeton University Press, Princeton, NJ, 2019.
  • Y. D. Sergeyev, Numerical point of view on Calculus for functions assuming finite, infinite and infinitesimal values over finite, infinite and infinitesimal domains, Nonlinear Anal., 71(12) (2009), 1688-1707.
  • H. Steinhaus, Mathematical Snapshots, 3rd Edition, Oxford University Press, New York, 1969.
Year 2022, Volume: 31 Issue: 31, 55 - 73, 17.01.2022
https://doi.org/10.24330/ieja.1058413

Abstract

References

  • W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann., 99(1) (1928), 118-133.
  • L. Antoniotti, F. Caldarola, G. d’Atri and M. Pellegrini, New approaches to basic calculus: an experimentation via numerical computation, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 329-342.
  • L. Antoniotti, F. Caldarola and M. Maiolo, Infinite numerical computing applied to Hilbert’s, Peano’s, and Moore’s curves, Mediterr. J. Math., 17(3) (2020), 99 (19 pp).
  • G. d’Atri, Logic-based consistency checking of XBRL instances, IJACT 3 (2014), 126-131.
  • A. A. Bennett, Note on an Operation of the Third Grade, Ann. of Math, Second Series, 17(2) (1915), 74-75.
  • J. Bowers, Exploding Array Function, Retrieved on 21-11-2018, http://www.polytope.net/hedrondude/array.htm.
  • F. Caldarola, The exact measures of the Sierpi´nski d-dimensional tetrahedron in connection with a Diophantine nonlinear system, Commun. Nonlinear Sci. and Numer. Simul., 63 (2018), 228-238.
  • F. Caldarola, The Sierpinski curve viewed by numerical computations with infinities and infinitesimals, Appl. Math. Comput., 318 (2018), 321-328.
  • F. Caldarola, D. Cortese, G. d’Atri and M. Maiolo, Paradoxes of the infinite and ontological dilemmas between ancient philosophy and modern mathematical solutions, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 358-372.
  • F. Caldarola, G. d’Atri, M. Maiolo and G. Pirillo, New algebraic and geometric constructs arising from Fibonacci numbers. In honor of Masami Ito, Soft Computing, 24(23) (2020), 17497-17508.
  • F. Caldarola, G. d’Atri, M. Maiolo and G. Pirillo, The sequence of Carboncettus octagons, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 373-380.
  • F. Caldarola, G. d’Atri, P. Mercuri and V. Talamanca, On the arithmetic of Knuth’s powers and some computational results about their density, In: Y.D. Sergeyev and D. Kvasov (eds.) Numerical Computations: Theory and Algorithms NUMTA 2019, Lecture Notes in Computer Science, vol. 11973 (2020), Springer, Cham, 381-388.
  • F. Caldarola and M. Maiolo, On the topological convergence of multi-rule sequences of sets and fractal patterns, Soft Computing, 24(23) (2020), 17737- 17749.
  • F. Caldarola, M. Maiolo and V. Solferino, A new approach to the Z-transform through infinite computation, Commun. Nonlinear Sci. Numer. Simul., 82 (2020), 105019.
  • J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, New York, 1957.
  • J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus, New York, 1996.
  • R. L. Goodstein, Transfinite ordinals in recursive number theory, J. Symbolic Logic, 12 (1947), 123-129.
  • R. L. Goodstein, On the restricted ordinal theorem, J. Symbolic Logic, 9 (1944), 33-41.
  • D. E. Knuth, Mathematics and computer Science: coping with finiteness, Science, 194(4271) (1976), 1235-1242.
  • A. Leonardis, Continued fractions in local fields and nested automorphisms, Ph.D. thesis, 2014.
  • R. Munafo, Inventing New Operators and Functions, Large Numbers at MROB, Retrieved on 19-11-2019.
  • R. Munafo, Versions of Ackermann’s Function, Large Numbers at MROB, Retrieved on 19-11-2019.
  • K. K. Nambiar, Ackermann Functions and Transfinite Ordinals, Appl. Math. Lett., 8(6) (1995), 51-53.
  • R. Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite, Princeton University Press, Princeton, NJ, 2019.
  • Y. D. Sergeyev, Numerical point of view on Calculus for functions assuming finite, infinite and infinitesimal values over finite, infinite and infinitesimal domains, Nonlinear Anal., 71(12) (2009), 1688-1707.
  • H. Steinhaus, Mathematical Snapshots, 3rd Edition, Oxford University Press, New York, 1969.
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Antonino Leonardıs This is me

Gianfranco D'atrı This is me

Fabio Caldarola This is me

Publication Date January 17, 2022
Published in Issue Year 2022 Volume: 31 Issue: 31

Cite

APA Leonardıs, A., D’atrı, G., & Caldarola, F. (2022). Beyond Knuth’s notation for unimaginable numbers within computational number theory. International Electronic Journal of Algebra, 31(31), 55-73. https://doi.org/10.24330/ieja.1058413
AMA Leonardıs A, D’atrı G, Caldarola F. Beyond Knuth’s notation for unimaginable numbers within computational number theory. IEJA. January 2022;31(31):55-73. doi:10.24330/ieja.1058413
Chicago Leonardıs, Antonino, Gianfranco D’atrı, and Fabio Caldarola. “Beyond Knuth’s Notation for Unimaginable Numbers Within Computational Number Theory”. International Electronic Journal of Algebra 31, no. 31 (January 2022): 55-73. https://doi.org/10.24330/ieja.1058413.
EndNote Leonardıs A, D’atrı G, Caldarola F (January 1, 2022) Beyond Knuth’s notation for unimaginable numbers within computational number theory. International Electronic Journal of Algebra 31 31 55–73.
IEEE A. Leonardıs, G. D’atrı, and F. Caldarola, “Beyond Knuth’s notation for unimaginable numbers within computational number theory”, IEJA, vol. 31, no. 31, pp. 55–73, 2022, doi: 10.24330/ieja.1058413.
ISNAD Leonardıs, Antonino et al. “Beyond Knuth’s Notation for Unimaginable Numbers Within Computational Number Theory”. International Electronic Journal of Algebra 31/31 (January 2022), 55-73. https://doi.org/10.24330/ieja.1058413.
JAMA Leonardıs A, D’atrı G, Caldarola F. Beyond Knuth’s notation for unimaginable numbers within computational number theory. IEJA. 2022;31:55–73.
MLA Leonardıs, Antonino et al. “Beyond Knuth’s Notation for Unimaginable Numbers Within Computational Number Theory”. International Electronic Journal of Algebra, vol. 31, no. 31, 2022, pp. 55-73, doi:10.24330/ieja.1058413.
Vancouver Leonardıs A, D’atrı G, Caldarola F. Beyond Knuth’s notation for unimaginable numbers within computational number theory. IEJA. 2022;31(31):55-73.