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A190356 Least positive x in the Diophantine equation x^3 + y^3 = n*z^3 (with x >= y and y != 0).

%I A190356 #49 Aug 05 2025 10:02:50
%S A190356 1,37,2,2,89,7,683,18,3,19,25469,3,3,163,137,1853,631,3,4,449,7,11,
%T A190356 23417,730511,1872,28747,5,11,4,4,5353,2538163,15409,53,197,17351,
%U A190356 5563,13,433,2570129,13,1176498611,53,1241,4,25903,15642626656646177,14,5,592,4033,165889,90,181,9109,5266097,5,184223499139,5,5,7,52954777
%N A190356 Least positive x in the Diophantine equation x^3 + y^3 = n*z^3 (with x >= y and y != 0).
%C A190356 This sequence a(k) is computed so that equation a(k)^3 + y^3 = A020898(k)*z^3 holds.
%C A190356 The 4 sequences A020898 [i.e., n], A190356 [i.e., x], A190580 [i.e., y] and A190581 [i.e., z] satisfy the equation A190356(n)^3 + A190580(n)^3 = A020898(n) * A190581(n)^3.
%C A190356 All x values above 25469 were obtained from Mishima's list and may not be the least positive solution.
%H A190356 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a Generalized Fermat-Wiles Equation</a> [broken link]
%H A190356 Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On a Generalized Fermat-Wiles Equation</a> [From the Wayback Machine]
%H A190356 Nakao Hisayasu, <a href="https://www.kaynet.or.jp/~kay/misc/nna2.html">Rational Points on Elliptic Curves: x^3+y^3=n</a> (nna2.html up to nna22.html)
%H A190356 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/x3y3s.txt">Solutions of Diophantine equation x^3+y^3=A.z^3 ...</a>
%e A190356 a(18) = 3 because A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.
%t A190356 (* Let x = u + v and y = u - v *)
%t A190356 f[n_, m_] := (r =  Reduce[u > 0 && v > 0 && Mod[2*u^3 + 6*u*v^2, n] == 0, {u, v},  Integers] ;
%t A190356 uv={u,v}/.(ToRules/@ List@@ r[[All,-2;;-1]])/.C-> c;
%t A190356 xy = (s = {};
%t A190356 Do[sel =  Select[uv,  IntegerQ[((2*#1[[1]]^3 + 6*#1[[1]]*#1[[2]]^2)/n)^(1/ 3)] &];
%t A190356 If[sel =!= {}, AppendTo[s, sel] ], {c[1], 0, m}, {c[2], 0,  m}];
%t A190356 {#[[1]] + #[[2]], #[[1]] - #[[2]]} & /@ (s //
%t A190356 Flatten[#, 1] &)) // Select[#, Total[#] != 0 &] &;
%t A190356 nxyz =  xy /. {x_Integer, y_} -> {n, x, y, ((x^3 + y^3)/n)^(1/3)};
%t A190356 nxyz /. ({_, x_, y_, z_} /; {x, y, z} != {0, 0, 0} &&
%t A190356 GCD[x, y, z] != 1) :> (gd = GCD[x, y, z]; {n, x/gd, y/gd, z/gd})) // Union // Sort[#, #1[[2]] < #2[[2]] &] &;
%t A190356 g[n_] := (m0 = 1; While[(r = f[n, m0]) == {}, m0 = 2 m0];
%t A190356 r // First);
%t A190356 A020898 = {2, 6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 49, 50, 51, 53, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 97, 98, 103, 105, 106, 107, 110, 114, 115, 117, 123, 124, 126, 127, 130}; km = Length[A020898]; (* xm(n) = some hard to compute values of x from Hisanori Mishima's list *) xm[22]=25469; xm[50]=23417; xm[51]=730511; xm[58]=28747; xm[68]=2538163; xm[69]=15409; xm[75]=17351; xm[85]=2570129; xm[87]=1176498611; xm[92]=25903; xm[94]=15642626656646177; xm[106]=165889; xm[114]=9109; xm[115]=5266097; xm[123]=184223499139; xm[130]=52954777; xm[n_] := xm[n] = g[n][[2]];
%t A190356 A190356 = Table[ n = A020898[[k]]; Print[xm[n]]; xm[n], {k, 1, km}] (* _Jean-François Alcover_, Jan 03 2012 *)
%Y A190356 Cf. A020898, A060838, A190580, A190581
%K A190356 nonn
%O A190356 1,2
%A A190356 _Jean-François Alcover_, May 11 2011
%E A190356 Positions corresponding to n=124 and n=127 (which were not minimal) corrected by _Jean-François Alcover_
%E A190356 Extended to 62 terms by _Jean-François Alcover_, Jan 03 2012