Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2014; 51(5): 1325-1337

Printed September 30, 2014

https://doi.org/10.4134/BKMS.2014.51.5.1325

Copyright © The Korean Mathematical Society.

On the last digit and the last non-zero digit of $n^n$ in base $b$

Jos\'{e} Mar\'{i}a Grau and Antonio M. Oller-Marc\'{e}n

Universidad de Oviedo, Centro Universitario de la Defensa

Abstract

In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base $b$. For the sequence given by the last digits of $n^n$ in base $b$, we prove its periodicity using different techniques than those used by W. Sierpinski and R. Hampel. In the case of the sequence given by the last non-zero digits of $n^n$ in base $b$ (which had been studied only for $b=10$) we show the non-periodicity of the sequence when $b$ is an odd prime power and when it is even and square-free. We also show that if $b=2^{2^s}$ the sequence is periodic and conjecture that this is the only such case.

Keywords: last digit, last non-zero digit, $n^n$

MSC numbers: 11B50

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