On the last digit and the last non-zero digit of \(n^n\) in base \(b\). (English) Zbl 1302.11002

Grau, José María; Oller-Marcén, Antonio M.

Bull. Korean Math. Soc. 51, No. 5, 1325-1337 (2014).

Summary: 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. Sierpiński [Ann. Soc. Polon. Math. 23, 252–258 (1950; Zbl 0040.30702)] and R. Hampel [Ann. Pol. Math. 1, 360–366 (1955; Zbl 0065.02804)]. 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.

MSC:

11A63	Radix representation; digital problems
11B50	Sequences (mod \(m\))

Citations:

Zbl 0040.30702; Zbl 0065.02804