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
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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