On the last digit and the last non-zero digit of $n^n$ in base $b$
Bull. Korean Math. Soc. 2014 Vol. 51, No. 5, 1325-1337
https://doi.org/10.4134/BKMS.2014.51.5.1325
Printed September 30, 2014
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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