A060467 Value of z 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, 1, 1, 1, 2, 2, 2, 2, 2, 3, -11, 2, 1626, 2, 3, 3, 3, 16, 15584139827, 3, 3, 3, 3, 3, 2220422932, 8866128975287528, 3, 3, 3, 4, 4, -159380, -80538738812075974, 3, 8, 4, 3, -8, 31, -796, -61922712865, 3, 12, 3, 22, 4, 5, 5, 3, 4, 4, 4, 4, 5, -21, 4, -10, 4

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) = 2220422932. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the last column of the table on Hisanori Mishima's web page. - M. F. Hasler, Nov 10 2015

Examples

			For n = 16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term 1626.
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, Section D5.

Links

• A. Bogomolny, Finicky Diophantine Equations on cut-the-knot.org, accessed Nov. 10, 2015.
• A.-S. Elsenhans, J. Jahnel, New sums of three cubes, Math. Comp. 78 (2009) 1227-1230.
• K. Koyama, Y. Tsuruoka, H. Sekigawa, On searching for solutions of the Diophantine equation x^3+y^3+z^3=n, Math. Comp. 66 (1997) 841.
• Hisanori Mishima, About n=x^3+y^3+z^3
• Eric S. Rowland, Known families of integer solutions of x^3+y^3+z^3=n
• A. Tyszka, A hypothetical upper bound for the solutions of a Diophantine equation with a finite number of solutions, arXiv:0901.2093 [math.NT], 2009-2014.

Crossrefs

Cf. A060464, A060465, A060466.

Programs

Mathematica

nmax = 29; A060464 = Select[Range[0, nmax], Mod[#, 9] != 4 && Mod[#, 9] != 5 &]; A060465 = {0, 0, 0, 1, -1, 0, 0, 0, 1, -2, 7, -1, -511, 1, -1, 0, 1, -11, -2901096694, -1, 0, 0, 0, 1}; r[n_, x_] := Reduce[0 <= Abs[x] <= Abs[y] <= Abs[z] && n == x^3 + y^3 + z^3, {y, z}, Integers]; A060467 = Table[z /. ToRules[ Simplify[ r[A060464[[k]], A060465[[k]]] /. C[1] -> 0]], {k, 1, Length[A060464]}] (* Jean-François Alcover, Jul 11 2012 *)

Extensions

In order to be consistent with A060465, where only primitive solutions are selected, a(18)=2 was replaced with 15584139827, by Jean-François Alcover, Jul 11 2012

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 14 2020

a(32) corrected by XU Pingya, May 11 2020