There are only four squares that are equal to the sum of three consecutive cubes.

Zero is one of them.

Find the other three.

DEFDBL A-Z
CLS
cu = 0: p = -1: pp = -8: sum = -9
FOR i = 1 TO 99999
 ppp = pp: pp = p: p = cu
 cu = i * i * i
 sum = sum + cu - ppp
 sr = INT(SQR(sum) + .5)
 IF sr * sr = sum THEN
   PRINT pp; p; cu, sum; sr
   PRINT i - 2; i - 1; i
   PRINT
 END IF
NEXT i

reports

-1  0  1                     0  0
-1  0  1

 0  1  8                     9  3
 0  1  2

 1  8  27                    36  6
 1  2  3

 12167  13824  15625         41616  204
 23  24  25
 

indicating the desired numbers are 0, 9, 36 and 41616.
 
 The first line of each set contains the three cubes to be added, followed by the resulting square and its square root.
 
 The second line in each contains the bases of the cubes, that is, the cube roots of the respective cubes.

Posted by Charlie on 2014-01-30 18:51:01