A060465 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|.

0, 0, 0, 1, -1, 0, 0, 0, 1, -2, 7, -1, -511, 1, -1, 0, 1, -11, -2901096694, -1, 0, 0, 0, 1, -283059965, -2736111468807040, -1, 0, 1, 0, 1, 117367, 12602123297335631, 2, -5, 2, -2, 6, -23, 602, 23961292454, -1, -7, 1, -11, 1, -1, 0, 2, 0, 0, 0, 1, 2, 11, -1, 7, 1

Offset: 0

Author

N. J. A. Sloane, Apr 10 2001

Keywords

• sign
• nice
• hard

Comments

Indexed by A060464.

Only primitive solutions where gcd(x,y,z) does not divide n are considered.

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

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

Examples

			For n = 16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term -511.
42 = 12602123297335631^3 + 80435758145817515^3 + (-80538738812075974)^3 was found by Andrew Booker and Andrew Sutherland.
74 = 66229832190556^3 + 283450105697727^3 + (-284650292555885)^3 was found by Sander Huisman.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, New York, 2004, Section D5, 231-234.

Links

• D. J. Bernstein, Three cubes
• A. Bogomolny, Finicky Diophantine Equations on cut-the-knot.org, accessed Nov. 10, 2015.
• B. Conn, L. Vaserstein, On sums of three integral cubes, Contemp. Math 166 (1994) MR1284068
• V. L. Gardiner, R. B. Lazarus, P. R. Stein, Solutions of the diophantine equation x^3+y^3=z^3-d, Math. Comp. 18 (1964) 408-413.
• D.R. Heath-Brown, W.M. Lioen and H.J.J. te Riele on Solving the Diophantine Equation x3 + y3 + z3 = k on a Vector Computer
• J. C. P. Miller, M. F. C. Woollett, Solutions of the Diophantine Equation x^3+y^3+z^3=k, J. Lond. Math. Soc. 30 (1) (1955) 101-110.
• Hisanori Mishima, About n=x^3+y^3+z^3

Crossrefs

Cf. A060464, A060466, A060467, A173515.

Programs

Mathematica

(* 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 *)

Extensions

Edited and a(24) added by M. F. Hasler, Nov 10 2015

a(25) from Tim Browning and further terms added by Charlie Neder, Mar 09 2019

More terms from Jinyuan Wang, Feb 13 2020