On Münchhausen numbers

Miroslav Kureš
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 1, Pages 14–21
DOI: 10.7546/nntdm.2021.27.1.14-21
Full paper (PDF, 266 Kb)

Details

Authors and affiliations

Miroslav Kureš
Department of Mathematics, Brno University of Technology
Technickа 2, 61669 Brno, Czechia

Abstract

The remarkable property of the number 3435, namely 3435 = 3³ + 4⁴ + 3³ + 5⁵, was formalized to the notion of Münchhausen numbers. Properties of these numbers are studied and it is showed that although there are only finitely many Münchhausen numbers in a given base b, there are infinitely many Münchhausen numbers of the length 2 in all bases. A certain reversion in the definition gives the notion of so-called anti-Munchhausen numbers, search for them is computationally more effective.

Keywords

• Münchhausen number
• Narcissistic number
• Sums of powers of integers

2010 Mathematics Subject Classification

• 11A67

References

1. van Berkel, D., On a curious property of3435, Available online at: https://arxiv.org/pdf/0911.3038v2.pdf
2. Elizarov, R.,MunchausenNumbers/output.txt, Available online at: https://github.com/elizarov/MunchausenNumbers/blob/master/output.txt