Sum of five consecutive cubes is a square

[Warut Roonguthai]

On 3/10/07, Nick Hobson <nickh at qbyte.org> wrote:

> Hi Seqfans,
>
> In the Dover (2005) edition of Dickson's History of the Theory of Numbers,
> Volume 2, Chapter 21, page 587, it is noted: "E. Lucas (275a) stated that
> the sum of the cubes of five consecutive integers is a square only when
> the middle number is 2, 3, 98 or 120."
> (275a) Recherches sur l'analyse indeterminee, Moulins, 1873, 92. Extract
>  from Bull. Soc. d'Emulation du Departement de l'Allier, 12, 1873, 532.
>
> The following PARI/GP script finds, additionally, middle number n = 27.
> Is this sequence finite?  Is such a short sequence (six terms, if we
> include n = 0) worth adding to OEIS?
>
> for(n=1, 10^8, if(issquare(5*n*(n*n+6)), print(n)))
>
> Nick

Yes, the sequence is finite because the number of integral points on an elliptic
curve is finite; in this case the curve is m^2 = 5n^3 + 30n.

Multiplying the equation by 25 and letting y = 5m and x = 5n, we have
y^2 = x^3 + 150x. According to Magma, the integral points on this curve are
(x, y) = (0, 0), (10, 50), (15, 75), (24, 132), (135, 1575), (490,
10850), (600, 14700).
So your list is complete. :)

Warut