1.Introduction

Problem of sum of consecutive squares equal to a square.
y^2 = (x+1)^2 + (x+2)^2 +....+ (x+n)^2.

2. Proposition

We consider below diophantine equation.

y^2 = nx^2+n(n+1)x+1/6n(n+1)(2n+1)...............................(1)


1. There exists infinity of integer solutions of equation (1)

Equation (1) is a quadratic diophantine equation, then we apply Gauss's theorem below.
Given one integer solution of equation (1) then there exists infinity of integer solutions 
If the conditions 1 and 2 are satisfied.

Condition 1: D=4n>0 and D is not perfect square.
Condition 2: discriminant=-1/3n^2(n-1)(n+1) is not zero.

Hence if n is perfect square, equation (1) has only finite solutions.


There is a curious solution such as [n,x,y]=[11, 37, 143],[33, 6, 143].

38^2 + 39^2 +...+ 48^2 = 7^2 + 8^2 +...+ 39^2 = 143^2

There is an another curious solution [n,x,y]=[11, 853, 2849],[74, 293, 2849].

Needless to say, equation (1) is related to Pell equation.


2. Equation (1) has no integer solution for n=8k+4,8k+5,8k+6.

Substitute n=8k+4 to RHS of equation (1).

RHS=(8k+4)x^2+(8k+4)(8k+5)x+1/6(8k+4)(8k+5)(16k+9) = 2 mod 4.

Thus equation y^2=2 mod 4 has no integer solution.

Hence Equation (1) has no integer solution for n=8k+4.
n=(4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92) with n<100.

Similary we can prove it for 8k+5 and 8k+6.
n=(5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93).
n=(6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94).




3.Numerical solutions

Search results of the integer points for equation (1) by brute force.
n<100
x<100000

[  n        x       y ]

[2, 2, 5]
[2, 19, 29]
[2, 118, 169]
[2, 695, 985]
[2, 4058, 5741]
[2, 23659, 33461]
[11, 17, 77]
[11, 37, 143]
[11, 455, 1529]
[11, 853, 2849]
[11, 9191, 30503]
[11, 17131, 56837]
[23, 6, 92]
[23, 16, 138]
[23, 880, 4278]
[23, 1350, 6532]
[23, 42786, 205252]
[23, 65336, 313398]
[24, 8, 106]
[24, 19, 158]
[24, 24, 182]
[24, 43, 274]
[24, 75, 430]
[24, 120, 650]
[24, 196, 1022]
[24, 303, 1546]
[24, 352, 1786]
[24, 539, 2702]
[24, 855, 4250]
[24, 1300, 6430]
[24, 2052, 10114]
[24, 3111, 15302]
[24, 3596, 17678]
[24, 5447, 26746]
[24, 8575, 42070]
[24, 12980, 63650]
[24, 20424, 100118]
[24, 30907, 151474]
[24, 35708, 174994]
[24, 54031, 264758]
[24, 84995, 416450]
[26, 24, 195]
[26, 300, 1599]
[26, 453, 2379]
[26, 3849, 19695]
[26, 31965, 163059]
[26, 47568, 242619]
[33, 6, 143]
[33, 26, 253]
[33, 59, 440]
[33, 180, 1133]
[33, 226, 1397]
[33, 611, 3608]
[33, 1084, 6325]
[33, 1984, 11495]
[33, 3491, 20152]
[33, 9046, 52063]
[33, 11160, 64207]
[33, 28859, 165880]
[33, 50606, 290807]
[33, 91986, 528517]
[47, 538, 3854]
[47, 730, 5170]
[47, 53930, 369890]
[47, 72358, 496226]
[49, 24, 357]
[50, 6, 245]
[50, 27, 385]
[50, 43, 495]
[50, 66, 655]
[50, 86, 795]
[50, 123, 1055]
[50, 167, 1365]
[50, 286, 2205]
[50, 378, 2855]
[50, 511, 3795]
[50, 627, 4615]
[50, 842, 6135]
[50, 1098, 7945]
[50, 1791, 12845]
[50, 2327, 16635]
[50, 3102, 22115]
[50, 3778, 26895]
[50, 5031, 35755]
[50, 6523, 46305]
[50, 10562, 74865]
[50, 13686, 96955]
[50, 18203, 128895]
[50, 22143, 156755]
[50, 29446, 208395]
[50, 38142, 269885]
[50, 61683, 436345]
[50, 79891, 565095]
[59, 21, 413]
[59, 37, 531]
[59, 1099, 8673]
[59, 1437, 11269]
[59, 55497, 426511]
[59, 72119, 554187]
[73, 441, 4088]
[73, 2697, 23360]
[74, 224, 2257]
[74, 293, 2849]
[74, 2513, 21941]
[74, 3184, 27713]
[74, 70265, 604765]
[74, 88760, 763865]
[88, 191, 2222]
[88, 224, 2530]
[88, 4151, 39358]
[88, 5924, 55990]
[88, 93011, 872938]
[96, 12, 652]
[96, 20, 724]
[96, 27, 788]
[96, 51, 1012]
[96, 136, 1828]
[96, 192, 2372]
[96, 215, 2596]
[96, 247, 2908]
[96, 580, 6164]
[96, 656, 6908]
[96, 723, 7564]
[96, 955, 9836]
[96, 1788, 17996]
[96, 2340, 23404]
[96, 2567, 25628]
[96, 2883, 28724]
[96, 6176, 60988]
[96, 6928, 68356]
[96, 7591, 74852]
[96, 9887, 97348]
[96, 18132, 178132]
[96, 23596, 231668]
[96, 25843, 253684]
[96, 28971, 284332]
[96, 61568, 603716]
[96, 69012, 676652]
[96, 75575, 740956]
[96, 98303, 963644]
[97, 14, 679]