A179407 - OEIS

%I #10 Jul 08 2015 04:07:07

%S 8,55,76,377,430,499,655,804,1827,5350,10805,15433,22108,44729,44817,

%T 96001,747343,748635,952463,7626590,10741787,12798893,14957531,

%U 15873532

%N 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).

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

%C For d values, see A179406.

%C For y values, see A179408.

%C Conjecture (from _Artur Jasinski_):

%C 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).

%H J. Blass, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0401638-2">A Note on Diophantine Equation Y^2 + k = X^5</a>, Math. Comp. 1976, Vol. 30, No. 135, pp. 638-640.

%H A. Bremner, <a href="http://dx.doi.org/10.1080/10586458.2008.10129039">On the Equation Y^2 = X^5 + k</a>, Experimental Mathematics 2008 Vol. 17, No. 3, pp. 371-374.

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

%t 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 *)

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

%K nonn,uned

%O 1,1

%A _Artur Jasinski_, Jul 13 2010