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%I #18 Sep 08 2023 22:37:29
%S 1,2,5,23,27,73,96,104,396,404,432,686,723,735,1130,1159,2019,2031,
%T 3861,5310,18219,18231,25592,25608,44367,200141,213842,308228,390615,
%U 390635,549976,631544,1579129,1657086,2941211,2941239,5523608
%N Values x for record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (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 A198443.
%C For y values see A198445.
%C Conjecture (_Artur Jasinski_): For any positive number x >= A198444(n), the distance d between the square of an integer y and the fifth power of x such that x <> k^2 and y <> k^5) can't be less than A198443(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)] + 1; k = m^2 - n^5; 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, 100000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; vecx
%Y Cf. A179406, A179407, A179408, A198443, A198445.
%K nonn,hard
%O 1,2
%A _Artur Jasinski_, Oct 25 2011
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