A051213 Numbers of the form 2^x-y^2 >= 0.

0, 1, 2, 3, 4, 7, 8, 12, 15, 16, 23, 28, 31, 32, 39, 47, 48, 55, 60, 63, 64, 71, 79, 87, 92, 103, 112, 119, 124, 127, 128, 135, 151, 156, 175, 183, 188, 192, 199, 207, 220, 223, 231, 240, 247, 252, 255, 256, 271, 284, 287, 295, 316, 343, 348, 367, 368, 375, 391, 399, 412, 431, 448

Offset: 1

Author

David W. Wilson

Keywords

• nonn

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

Crossrefs

Cf. A201125.

Programs

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 *)

PARI

is_A051213(n)=!A200522(n) \\ M. F. Hasler, Oct 09 2014

Extensions

More terms from M. F. Hasler, Oct 09 2014