Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a -sequence A satisfies . The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof of Erdős’ hypothesis for all even r=2k which we developped independently of Jia’s version.
@article{bwmeta1.element.bwnjournal-article-aav63i4p367bwm,
author = {Martin Helm},
title = {On $B\_{2k}$-sequences},
journal = {Acta Arithmetica},
volume = {64},
year = {1993},
pages = {367-371},
zbl = {0770.11010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav63i4p367bwm}
}
Martin Helm. On $B_{2k}$-sequences. Acta Arithmetica, Tome 64 (1993) pp. 367-371. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav63i4p367bwm/
[000] [1] H. Halberstam and K. F. Roth, Sequences, Springer, New York 1983.
[001] [2] X.-D. Jia, On B₆-sequences, J. Qufu Norm. Univ. Nat. Sci. 15 (3) (1989), 7-11.
[002] [3] X.-D. Jia, On -sequences, J. Number Theory, to appear.
[003] [4] J. C. M. Nash, On B₄-sequences , Canad. Math. Bull. 32 (1989), 446-449.
[004] [5] A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. II, J. Reine Angew. Math. 194 (1955), 111-140