A236210 Pairs of "harmonic numbers" 2^m * 3^n that differ by 1.

1, 2, 2, 3, 3, 4, 8, 9

Offset: 1

Author

Jonathan Sondow, Jan 20 2014

Keywords

• fini
• full
• nonn

Comments

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.

Examples

			8 + 1 = 2^3 + 1 = 3^2 = 9, so the pair 8 and 9 is in the sequence.
		

References

• L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.

Links

• A. Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234.
• H. Lenstra, Harmonic Numbers, MSRI, 1998.
• J. J. O'Connor and E. F. Robertson, Levi ben Gerson, The MacTutor History of Mathematics archive, 2009.
• I. Peterson, Medieval Harmony, Math Trek, MAA, 2012.
• Wikipedia, Gersonides
• Wikipedia, Philippe de Vitry

Crossrefs

Cf. A003586, A006899, A061987, A108906, A235365, A235366.