On the last digit and the last non-zero digit of nⁿ in base b

2012/03/19 by José María Grau, Grau, José María, Antonio M. Oller-Marcén +1
Mathematics · #11B50 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B50

paper · pdf · doi:10.48550/arxiv.1203.4066

arxiv created 2012/03/19 · arxiv updated 2012/03/20

Abstract

In this paper we study the sequences defined by the last and the last non-zero digits of nⁿ in base b. For the sequence given by the last digits of nⁿ 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 nⁿ 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=2^2s the sequence is periodic and conjecture that this is the only such case.

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