Problem of sum of consecutive squares equal to a cube. y^3 = (x+1)^2 + (x+2)^2 +....+ (x+n)^2.1.IntroductionWe consider below diophantine equation. y^3 = nx^2+n(n+1)x+1/6n(n+1)(2n+1)...............................(1)2. Proposition1. 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.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]3.Numerical solutions

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