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Balancing numbers which are products of three repdigits in base b

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Abstract

Repdigit in base b is a positive integer that has only one digit in its base b expansion, i.e. a number of the form \(a(b^m-1)/(b-1)\), for some positive integers \(m\ge 1\), \(b\ge 2\) and \(1\le a\le b-1\). In this paper, we investigate all balancing numbers which are expressed as products of three repdigits in base b. As a corollary, we show that the number 35 is the largest balancing number which can be expressible as a product of three repdigits. The proofs use Baker’s theory of lower bounds for nonzero linear forms in logarithms of algebraic numbers and reduction method due to Baker and Davenport, the Dujella-Pethő’s version.

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Acknowledgements

The author would like to express his gratitude to the anonymous referee for these various remarks which have qualitatively improved this paper. The author is supported by IMSP, Institut de Mathématiques et de Sciences Physiques de l’Université d’Abomey Calavi.

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  1. Institut de Mathématiques et de Sciences Physiques, Université d’Abomey, Calavi, Bénin

    Kouèssi Norbert Adédji

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  1. Kouèssi Norbert Adédji
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Adédji, K.N. Balancing numbers which are products of three repdigits in base b. Bol. Soc. Mat. Mex. 29, 45 (2023). https://doi.org/10.1007/s40590-023-00516-0

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