cp's OEIS Frontend

A179406 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).

%I A179406 #15 Sep 08 2023 22:39:39
%S A179406 7,19,60,341,47776,70378,78846,115775,220898,780231,2242100,11889984,
%T A179406 26914479,50406928,77146256,80117392,284679759,595974650,2071791247,
%U A179406 7825152599,67944824923,742629277177,1709838230002,2676465117663
%N A179406 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 A179406 Distance d is equal to 0 when x = k^2 and y = k^5.
%C A179406 For x values see A179407.
%C A179406 For y values see A179408.
%C A179406 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 A179406 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 A179406 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 A179406 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 A179406 Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408.
%K A179406 nonn,uned
%O A179406 1,1
%A A179406 _Artur Jasinski_, Jul 13 2010