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

(December 10, 2007) [Japanese]

### 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.

### References

 Richard K. Guy, Unsolved Problems in Number Theory (Second Edition), Springer, 1994.
 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.
 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.
 B. Conn and L. Vaserstein, On sums of three integral cubes, Contemp. Math. 166 (1994), 285-294.
 Noam D. Elkies, x^3 + y^3 + z^3 = d, NMBRTHRY archives (July 9, 1996)
 Eric Pine, Kim Yarbrough, Wayne Tarrant and Michael Beck, University of Georgia
 Noam D. Elkies,Rational points near curves and small nonzero |x3-y2| via lattice reduction,
ANTS IV (2000)
 D. J. Bernstein,
http://cr.yp.to/threecubes.html
 Leonid Durman,
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

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima