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%I #9 Oct 10 2016 02:45:46
%S 1,2,2,1,0,5,5,5,5,5,1,5,5,5,5,4,3,2,1,0,5,5,5,5,5,5,2,5,5,5,1,5,3,5,
%T 5,5,5,5,5,5,1,3,2,1,0,5,5,5,5,5,5,5,3,5,5,5,5,5,1,5,1,2,5,5,5,5,2,5,
%U 5,5,1,5,5,5,5,4,3,2,1,0,5,5,2,5,5,5,5,5,5,5,5,5,5,5,5,5,5,3,5,5,1,5,5,5,5
%N Smallest nonnegative value taken on by 5x^2 - ny^2 for an infinite number of integer pairs (x, y).
%t f[n_, z_] := FindInstance[x > 0 && y > 0 && 5*x^2 - n*y^2 == z, {x, y}, Integers, 1]; a[n_] := For[z = 0, z <= 5, z = z + GCD[n, 5], fz = f[n, z]; If[fz =!= {}, Print["a(", n, ") = ", z, " {x,y} = ", {x, y} /. fz[[1]]]; Return[z]]];
%t a[49] := (Print["a(49) = ", 5, " {x,y} = ", {51841, 16560}]; 5);
%t a[81] := (Print["a(81) = ", 5, " {x,y} = ", {161, 40}]; 5);
%t a[83] := (Print["a(83) = ", 5, " {x,y} = ", {1919, 471}]; 5);
%t a[98] := (Print["a(98) = ", 3, " {x,y} = ", {31, 7}]; 3);
%t Table[TimeConstrained[a[n], 300], {n, 1, 105}] (* _Jean-François Alcover_, Oct 10 2016 *)
%K nonn
%O 1,2
%A _David W. Wilson_
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