1.Introduction

I found that a12 could be expressed as a sum of three positive cubes by [a9(-1/3)] ways
in the Hardy's book.([1]. Hardy)

(9b4)3+(3a3b-9b4)3+(a4-9ab3)3=(a4)3

I actually calculated that Hardy said about above identity.

9(-1/3)=0.48075...

a=19 --> [a9(-1/3)]=9
a=21 --> [a9(-1/3)]=10
a=32 --> [a9(-1/3)]=15
a=42 --> [a9(-1/3)]=20                   
       

[1]. Hardy and Wright:  An introduction to the theory of numbers

2. Result


      1. 1303213 is expressed as a sum of three positive cubes by 9 ways.

     (19 1)   1303213 =       93+   205683+  1301503
     (19 2)           =     1443+   410103+  1289533
     (19 3)           =     7293+   610023+  1257043
     (19 4)           =    23043+   800043+  1193773
     (19 5)           =    56253+   972603+  1089463
     (19 6)           =   116643+  1117983+   933853
     (19 7)           =   216093+  1224303+   716683
     (19 8)           =   368643+  1277523+   427693
     (19 9)           =   590493+  1261443+    56623

      2. 1944813 is expressed as a sum of three positive cubes by 10 ways.

     (21 1)   1944813 =       93+   277743+  1942923
     (21 2)           =     1443+   554223+  1929693
     (21 3)           =     7293+   826203+  1893783
     (21 4)           =    23043+  1088283+  1823853
     (21 5)           =    56253+  1332903+  1708563
     (21 6)           =   116643+  1550343+  1536573
     (21 7)           =   216093+  1728723+  1296543
     (21 8)           =   368643+  1854003+   977133
     (21 9)           =   590493+  1909983+   567003
     (21 10)          =   900003+  1878303+    54813

      3. 10485763 is expressed as a sum of three positive cubes by 15 ways.

     (32 1)  10485763 =       93+   982953+ 10482883
     (32 2)           =     1443+  1964643+ 10462723
     (32 3)           =     7293+  2941833+ 10408003
     (32 4)           =    23043+  3909123+ 10301443
     (32 5)           =    56253+  4858953+ 10125763
     (32 6)           =   116643+  5781603+  9863683
     (32 7)           =   216093+  6665193+  9497923
     (32 8)           =   368643+  7495683+  9011203
     (32 9)           =   590493+  8256873+  8386243
     (32 10)          =   900003+  8930403+  7605763
     (32 11)          =  1317693+  9495753+  6652483
     (32 12)          =  1866243+  9930243+  5509123
     (32 13)          =  2570493+ 10209033+  4158403
     (32 14)          =  3457443+ 10305123+  2583043
     (32 15)          =  4556253+ 10189353+   765763

      4. 31116963 is expressed as a sum of three positive cubes by 20 ways.

     (42 1)  31116963 =       93+  2222553+ 31113183
     (42 2)           =     1443+  4443843+ 31086723
     (42 3)           =     7293+  6660633+ 31014903
     (42 4)           =    23043+  8867523+ 30875043
     (42 5)           =    56253+ 11056953+ 30644463
     (42 6)           =   116643+ 13219203+ 30300483
     (42 7)           =   216093+ 15342393+ 29820423
     (42 8)           =   368643+ 17412483+ 29181603
     (42 9)           =   590493+ 19413273+ 28361343
     (42 10)          =   900003+ 21326403+ 27336963
     (42 11)          =  1317693+ 23131353+ 26085783
     (42 12)          =  1866243+ 24805443+ 24585123
     (42 13)          =  2570493+ 26323833+ 22812303
     (42 14)          =  3457443+ 27659523+ 20744643
     (42 15)          =  4556253+ 28783353+ 18359463
     (42 16)          =  5898243+ 29664003+ 15634083
     (42 17)          =  7516893+ 30267993+ 12545823
     (42 18)          =  9447843+ 30559683+  9072003
     (42 19)          = 11728893+ 30501273+  5189943
     (42 20)          = 14400003+ 30052803+   876963































HOME