OFFSET
0,2
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
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.
FORMULA
a(2n) = a(3n) = 0.
a(n) <= 2 for n > 13.
EXAMPLE
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.
MATHEMATICA
Clear[seq]; seq[m_] := seq[m] = (Clear[a]; a[_] = 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 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jonathan Sondow, Dec 09 2012
EXTENSIONS
a(11) - a(30) from Robert Israel, Mar 06 2017
STATUS
approved