Chapter 4. n=x³+y³+z³ (D5 Sum of four cubes)

Abstract

In Unsolved Problem in Number Theory D5,

• Can arbitrary number n (≠9m±4) represent as a sum of three cubes,
  i.e. n=x³+y³+z³, x, y, z ∈ Z ?

• Can arbitrary number n represent as a sum of four cubes where two are same number,
  i.e. n=x³+y³+2z³, x, y, z ∈ Z ?

First we search the solutions of 1 ≤ n ≤ 10000 by simple search method.
Next, more refined way.

Contents

1. case 1 : n=x³+y³+z³
2. case 2 : n=x³+y³+2z³

References

[1] Richard K. Guy, Unsolved Problems in Number Theory (Second Edition), Springer, 1994.
[2] 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, Math. Comp. 61(1993),235-244.
[3] Kenji Koyama, Yukio Tsuruoka, and Hiroshi Sekigawa, On Searching for Solutions of the Diophantine Equation x³+y³+z³=n, Math. Comp. 55(1997),841-851.
[4] B. Conn and L. Vaserstein, On sums of three integral cubes, Contemp. Math. 166 (1994), 285-294.
[5] Noam D. Elkies, x^3 + y^3 + z^3 = d, NMBRTHRY archives (July 9, 1996)
[6] Eric Pine, Kim Yarbrough, Wayne Tarrant and Michael Beck, University of Georgia
[7] Noam D. Elkies,Rational points near curves and small nonzero |x³-y²| via lattice reduction,
ANTS IV (2000)
[8] D. J. Bernstein,
     http://cr.yp.to/threecubes.html
[9] Leonid Durman,
     http://www.uni-math.gwdg.de/jahnel/linkstopaperse.html
     http://www.uni-math.gwdg.de/jahnel/Arbeiten/Liste/threecubes_20070419.txt

Update

December 10, 2007

On April 19, 2007, Leonid Durman found the following results,

156 = 68844645625³ + 2232194323³ + (-68845427846)³
318 = 47835963799³ + 20549442727³ + (-49068024704)³
366 = 241832223257³ + 167734571306³ + (-266193616507)³
420 = 8859060149051³ + (-2680209928162)³ + (-8776520527687)³
564 = 53872419107³ + (-1300749634)³ + (-53872166335)³
758 = 662325744409³ + 109962567936³ + (-663334553003)³
789 = 18918117957926³ + 4836228687485³ + (-19022888796058)³
894 = 19868127639556³ + 2322626411251³ + (-19878702430997)³
933 = 998246159³ + (-165963535)³ + (-996714691)³
948 = 323019573172³ + 63657228055³ + (-323841549995)³

May 16, 2004

The team of University of Georgia had already found a solution of case n=52 for n=x³+y³+z³.
The solution is

52 = 60702901317³ + 23961292454³ + (-61922712865)³

May 04, 2003

Mike Oakes found;

3982 = -2 * 332669^3 - 11077811^3 + 11078011^3
5972 = 2 * 2980937^3 + 3002495^3 - 4309669^3
6124 = -2 * 209552^3 - 17513657^3 + 17513677^3
7151 = 2 * 3667541^3 + 3004190^3 - 5010331^3
8581 = -2 * 3842861^3 - 66333785^3 + 66342382^3
8653 = 2 * 30046^3 + 4252402^3 - 4252403^3

Now the remaining numbers are

148, 671, 788
1084, 1121, 1247, 1444, 1462, 1588, 1975
2246, 2300, 2372, 2822
3047, 3268, 3307, 3335, 3380, 3641, 3676, 3956
4036, 4108, 4369, 4388, 4819, 4883, 4990
5188, 5279, 5468, 5540, 5620, 5629
6707, 6980
7097, 7106, 7132, 7177, 7323, 7519, 7708, 7727, 7799, 7853, 7862, 7988
8114, 8380, 8572, 8588, 8644, 8779, 8887, 8968
9274, 9463, 9589, 9724, 9850

September 07, 2002

Both tables n=x³+y³+z³, n=x³+y³+2z³ are updated by D. J. Berstein and Jean-Charles Meyrignac.
The search range of n=x³+y³+z³ is up to 10^10 and beyond,
and the search range of n=x³+y³+2z³ is up to max(|x|,|y|,|z|) = 4000000.
(but unfortunately some data were missed.)

March 21, 2001

Search range of n=x³+y³+2z³ was enhanced from max { |x|, |y|, |z| } ≤ 700000 to 10⁶.
New results of this range are,

2176, 2561, 3730, 3784, 5908, 6548, 6782, 7276, 8104, 9320, 9526