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%I #15 Sep 08 2023 22:39:39
%S 7,19,60,341,47776,70378,78846,115775,220898,780231,2242100,11889984,
%T 26914479,50406928,77146256,80117392,284679759,595974650,2071791247,
%U 7825152599,67944824923,742629277177,1709838230002,2676465117663
%N Record minima of the positive distance d between the fifth power of a positive integer x and the square of an integer y such that 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 x values see A179407.
%C For y values see A179408.
%C 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).
%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.
%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}]; dd
%Y Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408.
%K nonn,uned
%O 1,1
%A _Artur Jasinski_, Jul 13 2010