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 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.2014.51.5.1325Printed 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 Downloads: Full-text PDF