A219551 Number of positive integer solutions to the equation |2^x - 3^y| = n.

0, 3, 0, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0

Offset: 0

Author

Jonathan Sondow, Dec 09 2012

Keywords

• nonn
• more

Comments

Pillai (1931) proved that a(n) is finite for all n.

Hershfeld (1936) computed a(n) for n <= 10 and proved that a(n) <= 2 for all large n.

Stroeker and Tijdeman (1982) proved that a(n) <= 2 for all n > 13.

For additional comments, references, and links, see the crossrefs.

a(n) <= 1 except for n=1, 5, 7, 13, 23: see e,g, Bennett (2003). - Robert Israel, Mar 06 2017

Examples

			1 = 2^2 - 3 = 3 - 2 = 3^2 - 2^3.
5 = 2^3 - 3 = 2^5 - 3^3 = 3^2 - 2^2.
7 = 2^4 - 3^2 = 3^2 - 2.
23 = 2^5 - 3^2 = 3^3 - 2^2 and a(n) <= 2 for n > 13, so a(23) = 2.
		

References

• S. Pillai, On the inequality 0 < a^x - b^y <= n, Journal Indian Math. Soc., 19 (1931), 1-11.
• R. J. Stroeker and R. Tijdeman, Diophantine equations, Computational methods in number theory, Part 2, Math. Cent. Tracts, 155 (1982), 321-369.

Links

• M. A. Bennett, On Some Exponential Equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
• M. A. Bennett, Pillai’s conjecture revisited, J. Number Theory, 98 (2003) 228-235.
• A. Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234.
• M. Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 2009.

Crossrefs

Cf. A053289, A074981, A076438, A207079.

Programs

Mathematica

Clear[seq]; seq[m_] := seq[m] = (Clear[a]; a[A219551%20=%20seq%5Bm%5D%20(*%20_Jean-Fran%C3%A7ois%20Alcover">] = 0; Do[n = Abs[2^x - 3^y]; a[n] = a[n] + 1, {x, 1, m}, {y, 1, m}]; Table[a[n], {n, 0, 10}]); seq[m = 1]; While[seq[m] != seq[m - 1], m = 2*m]; A219551 = seq[m] (* _Jean-François Alcover, Dec 13 2012 *)

Formula

a(2n) = a(3n) = 0.

a(n) <= 2 for n > 13.

Extensions

a(11) - a(30) from Robert Israel, Mar 06 2017