1.Introduction

It has been proven that all integers except 9m+4 and 9m-4 are representable by 
the sums of four cubes. ([1]. Demjanenko. )
Well, are all integers except 9m+4 and 9m-4 representable by the sums of three cubes?
n=X^3+Y^3+Z^3
The following integers cannot be solved within the range of n<1000 yet. 
n=33,42,74,114,165,390,579,627,633,732,795,906,921,975 


Recently(4/19/2007),solutions for n=156,318,366,420,564,758,789,894 and 948
are found by Elsenhans and Jahnel.([2].Elsenhans and Jahnel )


156= 26577110807569^3 -18161093358005^3 -23381515025762^3

318= 47835963799^3+ 20549442727^3 -49068024704^3
318= 1970320861387^3+ 1750553226136^3 -2352152467181^3
318= 30828727881037^3+ 27378037791169^3 -36796384363814^3

366= 241832223257^3+ 167734571306^3 -266193616507^3

420= 8859060149051^3 -2680209928162^3 -8776520527687^3

564= 53872419107^3 -1300749634^3 -53872166335^3

758= 662325744409^3+ 109962567936^3 -663334553003^3

789= 18918117957926^3+ 4836228687485^3 -19022888796058^3

894= 19868127639556^3+ 2322626411251^3 -19878702430997^3

948= 323019573172^3+ 63657228055^3 -323841549995^3
948= 103458528103519^3+ 6604706697037^3 -103467499687004^3




For n>1000,there are few known results, and they are old.
So, I searched for the solutions of n=X^3+Y^3+Z^3 which the no solutions were known.
I found 184 solutions , but it seems that the search range must be expanded.

[1/8/2008]
n=2035,2895,3323,3594,3764,4629,5079,5091,5196,5596,6267,6873,7086,7116,7478,
7683,7788,7917,8355,and 8754 were added to the list.

[1/15/2008]
n=1263,1361,2931,3201,3471,4764,5628,6123,6501,6513,6708,7518,7815,8062,8530,
8628,8673,8979,9691,9726 were added to the list.

[2/7/2008]
n=1131,1965,2613 and 2923 were added to the list.

[1].Demjanenko:http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps 
[2].Elsenhans and Jahnel:http://www.uni-math.gwdg.de/jahnel/Preprints/elk_ants6c.pdf      
                        :http://www.uni-math.gwdg.de/jahnel/Arbeiten/Liste/threecubes_20070419.txt





2.Search results

      Min(X,Y,Z) < 10^9

a. 1000<=|d|<=1999

b. 2000<=|d|<=2999

c. 3000<=|d|<=3999

d. 4000<=|d|<=4999

e. 5000<=|d|<=5999

f. 6000<=|d|<=6999

g. 7000<=|d|<=7999

h. 8000<=|d|<=8999

i. 9000<=|d|<=9999