---------- Forwarded message ---------- Date: Sat, 28 Dec 2002 12:06:13 -0500 From: Robert Harley <harley@argote.ch> To: NMBRTHRY@LISTSERV.NODAK.EDU Subject: Re: Integers with many 0s in base 2 and base 3 Andreas Weingartner wrote:

Prove or disprove: There exists an epsilon>0, such that no natural number has the property that in base 2 as well as in base 3, at most (epsilon)*100% of the digits are nonzero.

Heuristically one might expect (except possibly for some small examples) such an eps for bases a and b at the solution of: (2*eps-1)*log(a)*log(b) = (eps*log(eps)+(1-eps)*log(1-eps))*(log(a)+log(b)) For bases 2 and 3, this is about 0.104939... I ran a search up to 2^64 for small examples: 6: 0.666667 9: 0.5 18: 0.4 162: 0.375 261: 0.333333 4376: 0.307692 19712: 0.3 32805: 0.25 65610: 0.235294 131220: 0.222222 4785156: 0.217391 9570312: 0.208333 272629962: 0.206897 1208614932: 0.2 2542645806624: 0.190476 154206526918656: 0.1875 2348694485729280: 0.181818 9341451062288388: 0.176471 18049789376104448: 0.171429 451521135633235968: 0.169492 Regards, Rob. .-. .-. / \ .-. .-. / \ / \ / \ .-. _ .-. / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / `-' `-' \ / \ / \ \ / `-' `-' \ / `-' `-'