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%I #12 Mar 28 2015 17:10:16
%S 1,2,2,3,3,4,8,9
%N Pairs of "harmonic numbers" 2^m * 3^n that differ by 1.
%C Philippe de Vitry (1291-1361), a musician from Vitry-en-Artois in France, called numbers of the form 2^m * 3^n "harmonic numbers". He asked if all powers of 2 and 3 differ by more than 1 except the pairs 1 and 2, 2 and 3, 3 and 4, 8 and 9 (which correspond to musically significant ratios, representing an octave, fifth, fourth, and whole tone). Levi Ben Gerson (1288-1344) answered yes by proving that 3^n +- 1 is not a power of 2 if n > 2; see A235365, A235366.
%D L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.
%H A. Herschfeld, <a href="http://dx.doi.org/10.1090/S0002-9904-1936-06275-0">The equation 2^x - 3^y = d</a>, Bull. Amer. Math. Soc., 42 (1936), 231-234.
%H H. Lenstra, <a href="http://www.msri.org/publications/ln/msri/1998/mandm/lenstra/1/index.html">Harmonic Numbers</a>, MSRI, 1998.
%H J. J. O'Connor and E. F. Robertson, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Levi.html">Levi ben Gerson</a>, The MacTutor History of Mathematics archive, 2009.
%H I. Peterson, <a href="http://archive.is/iRXz">Medieval Harmony</a>, Math Trek, MAA, 2012.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gersonides">Gersonides</a>
%H Wikipedia, <a href="http://en.m.wikipedia.org/wiki/Philippe_de_Vitry">Philippe de Vitry</a>
%e 8 + 1 = 2^3 + 1 = 3^2 = 9, so the pair 8 and 9 is in the sequence.
%Y Cf. A003586, A006899, A061987, A108906, A235365, A235366.
%K fini,full,nonn
%O 1,2
%A _Jonathan Sondow_, Jan 20 2014
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