|
%I #10 Jul 08 2015 04:07:16
%S 181,22434,50354,2759646,3834168,5562261,10980023,18329057,142674503,
%T 2093555387,12135618855,29588700403,72673092233,423129175811,
%U 425213412449,2855547523353,482836315990072,484925830443335
%N Values y for records of minima of positive distance d between a fifth power of a positive integer x and a square of an 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 x values, see A179407.
%C Conjecture (from _Artur Jasinski_):
%C For any positive number x >= A179407(n), the distance d between 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 A179407(n)^5-a(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}]; yy (* _Artur Jasinski_, Jul 13 2010 *)
%Y Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179406, A179407.
%K nonn,uned
%O 1,1
%A _Artur Jasinski_, Jul 13 2010
|
