1.Introduction

Problem of sum of consecutive cubes equal to a perfect square.
y^2 = x^3 + (x+1)^3 +....+ (x+n-1)^3.
R. J. Stroeker[1] showed the solutions this problem with n<50.
We extened the solution range to n<100 and showed the partial solutions for even values of n.

2. Claim

We consider below diophantine equation.

y^2 = nx^3+3/2n(n-1)x^2+1/2n(n-1)(2n-1)x+1/4n^2(n-1)^2.................(1)

Equation (1) is birationally equivalent to an elliptic curve below.

Y^2 = X^3 + 1/4n^2(n^2-1)X.............................................(2)

X=nx+n(n-1)/2, Y=ny.

1. Equation (1) always has the integer solutions for odd values of n

We know equation (2) has a solution (X,Y) = ( n^2(n-1)(n+1), 1/2n^2(n-1)(n+1)(2n^2-1) ).

Hence we obtain (x,y)=( 1/2(n-1)(2n^2+2n-1), 1/2n(n-1)(n+1)(2n^2-1) ).

Thus (x,y) is always the integer for odd values of n.

2. Equation (1) has infinitely many integer solutions for even values of n=2m^2

Let n=2m^2, then equation (2) has a solution (X,Y) = (4m^6-m^2, 2m^5(2m^2-1)(2m^2+1)).

Hence we obtain (x,y)=( m^2(2m^2-1), m^3(2m^2-1)(2m^2+1) ).
m is arbitrary.

m<20

[  n      x          y ]
[  8      28        504]
[ 18     153       8721]
[ 32     496      65472]
[ 50    1225     312375]
[ 72    2556    1119528]
[ 98    4753    3293829]
[128    8128    8388096]
[162   13041   19131147]
[200   19900   39999000]
[242   29161   77947353]
[288   41328  143325504]
[338   56953  250991871]
[392   76636  421651272]
[450  101025  683434125]
[512  130816 1073737728]
[578  166753 1641349779]
[648  209628 2448874296]
[722  260281 3575480097]




3.Numerical solutions

Search results of the integer points for equation (1) using Online Magma Calculator.

[  n         x          y ]

[  3         23        204]
[  5         25        315]
[  5         96       2170]
[  5        118       2940]
[  7        333      16296]
[  8         28        504]
[  9        716      57960]
[ 11       1315     159060]
[ 12         14        312]
[ 13        144       6630]
[ 13       2178     368004]
[ 15         25        720]
[ 15       3353     754320]
[ 15      57960   54052635]
[ 17          9        323]
[ 17        120       5984]
[ 17       4888    1412496]
[ 18        153       8721]
[ 18        680      76653]
[ 19       6831    2465820]
[ 21         14        588]
[ 21        144       8778]
[ 21       9230    4070220]
[ 23      12133    6418104]
[ 25      15588    9742200]
[ 27      19643   14319396]
[ 28         81       4914]
[ 29      24346   20474580]
[ 31      29745   28584480]
[ 32         69       4472]
[ 32        133      10296]
[ 32        496      65472]
[ 33         33       2079]
[ 33      35888   39081504]
[ 35        225      22330]
[ 35      42823   52457580]
[ 37      50598   69267996]
[ 39        111       9360]
[ 39      59261   90135240]
[ 40       3276    1196520]
[ 41      68860  115752840]
[ 42         64       5187]
[ 43      79443  146889204]
[ 45        176      18810]
[ 45      91058  184391460]
[ 47     103753  229189296]
[ 48         64       5880]
[ 48        410      62628]
[ 48      19881   19455744]
[ 48      60040  101985072]
[ 49     117576  282298800]
[ 50       1225     312375]
[ 51     132575  344826300]
[ 53     148798  417972204]
[ 54        265      36729]
[ 54       1272     343917]
[ 55     166293  503034840]
[ 57       1625     507471]
[ 57     185108  601414296]
[ 59     205291  714616260]
[ 60        118      14160]
[ 61     226890  844255860]
[ 63        217      31248]
[ 63        837     203112]
[ 63       1121     310464]
[ 63     249953  992061504]
[ 64        105      13104]
[ 64      34272   50827392]
[ 65     274528 1159878720]
[ 67     300663 1349673996]
[ 69         81      10695]
[ 69     328406 1563538620]
[ 71     357805 1803692520]
[ 72       2556    1119528]
[ 73      16864   18771220]
[ 73      26937   37849113]
[ 73     388908 2072488104]
[ 75     421763 2372414100]
[ 76        295      53200]
[ 77        144      22022]
[ 77     456418 2706099396]
[ 79     492921 3076316880]
[ 81     531320 3485987280]
[ 82        144      23247]
[ 83     571663 3938183004]
[ 85     613998 4436131980]
[ 87        232      43065]
[ 87     658373 4983221496]
[ 89     704836 5583002040]
[ 91       4785    3202290]
[ 91     753435 6239191140]
[ 92       4992    3429530]
[ 93     804218 6955677204]
[ 94        400      91979]
[ 95     857233 7736523360]
[ 97      23668   35970704]
[ 97      33660   60951696]
[ 97     912528 8585971296]
[ 98         25       7497]
[ 98         97      18333]
[ 98        216      43309]
[ 98        745     221697]
[ 98        760     227997]
[ 98       3961    2513511]
[ 98       4753    3293829]
[ 99     970151 9508445100]

4.Reference

[1]: R. J. Stroeker, On the sum of consecutive cubes being a square. Compos. Math. 97 (1995), 295–307.