OFFSET
1,3
COMMENTS
Is 519 in this sequence? Then this is the value of a(73), else it is 527, after which the sequence goes on with 540, 583, 604, 615, 623, 624,... - M. F. Hasler, Oct 09 2014
From R. J. Mathar, Oct 21 2014: (Start)
519 is not in the sequence. [Proof: Consider 2^x-519=y^2 and both sides modulo 3.
Then 2^x-519 = 1,2,1,2.... (mod 3) for x>=0 and y^2=0,1,1,0,1,1,... (mod 3) for y>=0.
For moduli to match (i.e, both 1), x must be even. Then 2^x is the square of the integer y=2^(x/2). (Note that this reference does not work in integers if x is odd).
The next smaller perfect square is (y-1)^2 = (2^(x/2)-1)^2 = 2^x-2^(1+x/2)+1 .
This must be >=2^x-519 to have a solution, so -2^(1+x/2)+1 >= -519
implies 2^(1+x/2)-1 <= 519, which implies 1+x/2 <= 9.02 and x<=16.
One can check numerically that the range 0<=x<=16 do not form perfect squares 2^x-519.] (End)
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..72
J. Cohn, The diophantine equation x^2+C=y^n, Acta Arithm. 65 (4) (1993) 367-381
Fadwa S. Abu Muriefah, Yann Bugeaud, The diophantine equation x^2+c=y^n: a brief overview, Rev. Colomb. Matem. 40 (1) (2006) 31-37
MATHEMATICA
max = 1000; Clear[f]; f[m_] := f[m] = Select[Table[2^x - y^2, {x, 0, m}, {y, 0, Ceiling[2^(x/2)]}] // Flatten // Union, 0 <= # <= max &]; f[1]; f[m = 2]; While[f[m] != f[m - 1], m++]; Print["m = ", m]; A051213 = f[m] (* Jean-François Alcover, May 13 2017 *)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from M. F. Hasler, Oct 09 2014
STATUS
approved