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On the number of binary digits in a multiple of three. (English) Zbl 0194.35004
L. Moser conjectured that considering the binary expansion of multiples of three $$(11, 110, 1001, \dots)$$ we can always find a preponderance of numbers containing an even number of digits 1 over those containing an odd number of digits 1. I. Barrodale and R. MacLeod verified this up to 500,000. This article contains a proof of the mentioned conjecture: Denote $$D(n)$$ the number of digits 1 in the binary expansion of $$n$$ and $S(N)=\sum_{0\leq j\leq N/3} (-1)^{D(N-3j)}.$ Theorem: For arbitrary $$n$$ $1/20<S(3n)/n^\alpha<5, \quad\text{where}\;\alpha=\log 3/\log 4.$ From the first inequality the Moser’s inequality follows. Further it is proved that $\lim_{n\to\infty} (S(3n)/n^\alpha)$ does not exist. The proof uses analytical methods but it is noted that S. Klein has found an elementary proof.
Reviewer: Št. Znám