A179407 Values x for records of minima of positive distance d between a fifth power of positive integer x and a square of integer y such d = x^5 - y^2 (x != k^2 and y != k^5).

8, 55, 76, 377, 430, 499, 655, 804, 1827, 5350, 10805, 15433, 22108, 44729, 44817, 96001, 747343, 748635, 952463, 7626590, 10741787, 12798893, 14957531, 15873532

Offset: 1

Author

Artur Jasinski, Jul 13 2010

Keywords

• nonn
• uned

Comments

Distance d is equal to 0 when x = k^2 and y = k^5.

For d values, see A179406.

For y values, see A179408.

Conjecture (from Artur Jasinski):

For any positive number x >= A179407(n), the distance d between the fifth power of x and the square of any y (such that x != k^2 and y != k^5) can't be less than A179406(n).

Links

• J. Blass, A Note on Diophantine Equation Y^2 + k = X^5, Math. Comp. 1976, Vol. 30, No. 135, pp. 638-640.
• A. Bremner, On the Equation Y^2 = X^5 + k, Experimental Mathematics 2008 Vol. 17, No. 3, pp. 371-374.

Crossrefs

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179406, A179408.

Programs

Mathematica

max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^5)^(1/2)]; k = n^5 - m^2; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 96001}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; xx (* Artur Jasinski, Jul 13 2010 *)

Formula

a(n)^5-A179408(n)^2 = A179406(n).