A060466 Value of y 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, 1, 1, -1, -1, 0, 1, 1, -2, 10, 2, -1609, 2, -2, -2, -2, -14, -15550555555, -1, -1, 0, 1, 1, -2218888517, -8778405442862239, 2, 2, 2, -3, -3, 134476, 80435758145817515, 2, -7, -3, 3, 7, -26, 659, 60702901317, 3, -11, 3, -21, -2, -4, -4, 3, -1, 0, 1, 1, -4, 20, 2, 9, 2

Offset

0,10

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) = -2218888517. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the second column of the table on Hisanori Mishima's web page. - M. F. Hasler, Nov 10 2015

References

R. K. Guy, Unsolved Problems in Number Theory, Section D5.

Links

Table of n, a(n) for n=0..57.

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.

Example

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

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]; A060466 = Table[y /. ToRules[ Simplify[ r[A060464[[k]], A060465[[k]]] /. C[1] -> 0]], {k, 1, Length[A060464]}] (* Jean-François Alcover, Jul 11 2012 *)

Crossrefs

Cf. A060465, A060467, A173515.

Sequence in context: A246479 A359694 A171659 * A243992 A317549 A337321

Adjacent sequences: A060463 A060464 A060465 * A060467 A060468 A060469

Keyword

sign,nice,hard

Author

N. J. A. Sloane, Apr 10 2001

Extensions

In order to be consistent with A060465, where only primitive solutions are selected, a(18)=2 was replaced with -15550555555, 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

Status

approved