1.Introduction

Problem of sum of consecutive squares equal to a cube.
y^3 = (x+1)^2 + (x+2)^2 +....+ (x+n)^2.

2. Proposition

We consider below diophantine equation.

y^3 = nx^2+n(n+1)x+1/6n(n+1)(2n+1)...............................(1)

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

Since discriminant for x must be perfect square number, then we obtain

v^2 = -3n^4+3n^2+36y^3n..........................................(2)

We consider equation (2) as quartic equation of n and searched the integer solutions of equation (2).

Let y=n, then v^2 = 3n^2(11n^2+1).

Hence we consider the integer solution of 11n^2+1 = 3u^2.........(3)

Since equation (3) has a solution (u,n)=(2,1), then it has infinitely many integer solutions as follows.

Obviously, equation (3) is Pell equation.

Integer solutions are given below recursive sequence.

(x0,y0)=(2,1)
x(i+1) = 23 x(i) + 44 y(i) 
y(i+1) = 12 x(i) + 23 y(i) 

            [n   x   y ]
       [1], [47, 21, 47]
       [2], [2161, 988, 2161]
       [3], [99359, 45449, 99359]
       [4], [4568353, 2089688, 4568353]
       [5], [210044879, 96080221, 210044879]
       [6], [9657496081, 4417600500, 9657496081]
       [7], [444034774847, 203113542801, 444034774847]
       [8], [20415942146881, 9338805368368, 20415942146881]
       [9], [938689303981679, 429381933402149, 938689303981679]


Hence there exists infinity of integer solutions of equation (1).


2. Equation (1) has no solution for n=27k+3, 27k+9, 27k+18, 27k+24.

Substitute n=27k+3 to RHS of equation (1).

RHS = 3x+3x^2+5 = 2,5 mod 9.

Thus equation y^3 = 2,5 mod 9 has no solution.

Hence Equation (1) has no solution for n=27k+3.

Similary we can prove it for 27k+9, 27k+18, 27k+24.




3.Numerical solutions

Search results of the integer points for equation (1) by brute force.

n<1000000
y<1000000

[  n    x    y ]

[26, 89, 65]
[47, 21, 47]
[162, 2114, 921]
[393, 125089, 18340]
[2161, 988, 2161]
[3071, 23016, 12284]
[3807, 260, 2820]
[4307, 248057, 64605]
[5577, 17353, 13156]
[5994, 24977, 16761]
[7802, 189431, 66317]
[10368, 83867, 43500]
[21531, 279607, 122009]
[24649, 239805, 116180]
[34991, 927207, 314919]
[36842, 64790, 63765]
[57967, 1033992, 403130]
[65522, 24078, 61721]
[99359, 45449, 99359]
[101306, 2312936, 827209]
[123877, 1015470, 524095]
[137842, 77288, 146821]
[148877, 84754, 159371]
[156099, 254785, 260165]
[170368, 250353, 269588]
[528000, 733891, 813340]
[572427, 772113, 869241]
[674816, 838167, 985864]
[942841, 52910, 690381]