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.txt1.IntroductionMin(X,Y,Z) < 10^92.Search resultsa. 1000<=|d|<=1999b. 2000<=|d|<=2999c. 3000<=|d|<=3999d. 4000<=|d|<=4999e. 5000<=|d|<=5999f. 6000<=|d|<=6999g. 7000<=|d|<=7999h. 8000<=|d|<=8999i. 9000<=|d|<=9999

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