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%I #50 Mar 04 2020 02:22:26
%S 0,0,0,1,-1,0,0,0,1,-2,7,-1,-511,1,-1,0,1,-11,-2901096694,-1,0,0,0,1,
%T -283059965,-2736111468807040,-1,0,1,0,1,117367,12602123297335631,2,
%U -5,2,-2,6,-23,602,23961292454,-1,-7,1,-11,1,-1,0,2,0,0,0,1,2,11,-1,7,1
%N Value of x of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.
%C Indexed by A060464.
%C Only primitive solutions where gcd(x,y,z) does not divide n are considered.
%C From the solution A060464(24) = 30 = -283059965^3 - 2218888517^3 + 2220422932^3 (smallest possible magnitudes according to A. Bogomolny), one has a(24) = -283059965. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the first column of the table on Hisanori Mishima's web page. - _M. F. Hasler_, Nov 10 2015
%C In 2019 Brooker found a solution for n = 33 (see A332201 and references there) and later in the same year for n = 42, using the collaborative "Charity Engine". It would be nice to have information on how far it is established that these solutions are the smallest possible. - _M. F. Hasler_, Feb 24 2020
%D R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, New York, 2004, Section D5, 231-234.
%H D. J. Bernstein, <a href="http://cr.yp.to/threecubes.html">Three cubes</a>
%H A. Bogomolny, <a href="http://www.cut-the-knot.org/arithmetic/algebra/FinikyDiophantineEquations.shtml">Finicky Diophantine Equations</a> on cut-the-knot.org, accessed Nov. 10, 2015.
%H B. Conn, L. Vaserstein, <a href="https://bookstore.ams.org/conm-166">On sums of three integral cubes</a>, Contemp. Math 166 (1994) <a href="http://www.ams.org/mathscinet-getitem?mr=1284068">MR1284068</a>
%H V. L. Gardiner, R. B. Lazarus, P. R. Stein, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0175843-9">Solutions of the diophantine equation x^3+y^3=z^3-d</a>, Math. Comp. 18 (1964) 408-413.
%H D.R. Heath-Brown, W.M. Lioen and H.J.J. te Riele <a href="http://euler.free.fr/docs/HLR93.pdf"> on Solving the Diophantine Equation x3 + y3 + z3 = k on a Vector Computer</a>
%H J. C. P. Miller, M. F. C. Woollett, <a href="http://dx.doi.org/10.1112/jlms/s1-30.1.101">Solutions of the Diophantine Equation x^3+y^3+z^3=k</a>, J. Lond. Math. Soc. 30 (1) (1955) 101-110.
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm">About n=x^3+y^3+z^3</a>
%e For n = 16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term -511.
%e 42 = 12602123297335631^3 + 80435758145817515^3 + (-80538738812075974)^3 was found by Andrew Booker and Andrew Sutherland.
%e 74 = 66229832190556^3 + 283450105697727^3 + (-284650292555885)^3 was found by Sander Huisman.
%t (* this program is not convenient for hard cases *) nmax = 29; xmin[_] = 0; xmax[_] = 20; xmin[16] = 500; xmax[16] = 600; xmin[24] = 2901096600; xmax[24] = 2901096700; r[n_, x_] := Reduce[0 <= Abs[x] <= Abs[y] <= Abs[z] && n == x^3 + y^3 + z^3, {y, z}, Integers]; r[n_ /; IntegerQ[n^(1/3)]] := {0, 0, n^(1/3)}; mySort = Sort[#1, Which[Abs[#1[[3]]] <= Abs[#2[[3]]], True, Abs[#1[[3]]] == Abs[#2[[3]]], If[Abs[#1[[2]]] <= Abs[#2[[2]]], True, False], True, False] & ] & ; rep := {x_, y_, z_} /; (x + y == 0 && x > 0) :> {-x, -y, z}; r[n_] := Reap[Do[ sp = r[n, x] /. C[1] -> 1; If[sp =!= False, xyz = {x, y, z} /. {ToRules[sp]} /. rep; If[GCD @@ Flatten[{n, xyz}] == 1, Sow[xyz]]]; sn = r[n, -x] /. C[1] -> 1; If[sn =!= False, xyz = {-x, y, z} /. {ToRules[sn]} /. rep; If[GCD @@ Flatten[{n, xyz}] == 1, Sow[xyz]]], {x, xmin[n], xmax[n]}]][[2, 1]] // Flatten[#, 1] & // mySort // First; A060464 = Select[Range[0, nmax], Mod[#, 9] != 4 && Mod[#, 9] != 5 &]; A060465 = Table[xyz = r[n]; Print[ " n = ", n, " {x,y,z} = ", xyz]; xyz[[1]], {n, A060464}] (* _Jean-François Alcover_, Jul 10 2012 *)
%Y Cf. A060464, A060466, A060467, A173515.
%K sign,nice,hard
%O 0,10
%A _N. J. A. Sloane_, Apr 10 2001
%E Edited and a(24) added by _M. F. Hasler_, Nov 10 2015
%E a(25) from Tim Browning and further terms added by _Charlie Neder_, Mar 09 2019
%E More terms from _Jinyuan Wang_, Feb 13 2020
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