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Zbl 0837.11012
Stroeker, R.J.
On the sum of consecutive cubes being a perfect square. (English)
[J] Compos. Math. 97, No.1-2, 295-307 (1995). ISSN 0010-437X; ISSN 1570-5846/e

In this paper, the author describes a systematic method that answers the following problem: Given $n \in \bbfN$, find all integers $x$ for which $x^3 + (x + 1)^3 + \cdots + (x + n - 1)^3$ is a perfect square. Clearly, for $x = 1$ and any $n$, this sum is a perfect square but, if one is interested in exhausting all such $x$ (for a given $n)$ he can find only sparse results in the literature and, surely, no result of a general character.\par The author makes the observation that the above problem can be answered if one can explicitly calculate all integral points on the (already extensively investigated) elliptic curve $Y^2 = X^3 + d_n \cdot X$, where $d_n = n^2(n^2 - 1)/4$. Recently, the author and the reviewer elaborated a method for finding all integral points on a Weierstrass model of an elliptic curve [Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)]. Independently, {\it J. Gebel}, {\it A. Pethö} and {\it H. Zimmer} developed a similar method [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)]. The realization of this method is heavily based on a recent estimate of {\it S. David} for linear forms in elliptic logarithms [Mém. Soc. Math. Fr. (to appear)].\par The advantage of the method is that, once the generators for the Mordell- Weil group of the corresponding elliptic curve are known, a number of clear steps, independent of any ad hoc arguments, lead to the explicit determination of all integral points. On the other hand, its uniform character permits one to deal with many curves simultaneously. The author takes advantage of this feature and utilizes succesfully the method in order to answer explicitly to the initial problem for all $n$ from 2 to 50 and for $n = 98$. His strategy, in its essential lines, is independent from those particular values of $n$ and, surely, can be applied to other values of $n$ as well. The paper is very neatly written and its style is attractive.
[N.Tzanakis (Iraklion)]

MSC 2000:: *11D25 Cubic and quartic diophantine equations
11J86 Linear forms in logarithms; Baker's method
11D85 Representation problems of integers

Keywords: sum of consecutive cubes; integral points of an elliptic curve; LLL- reduction; perfect square; linear forms in elliptic logarithms

Citations: Zbl 0805.11026; Zbl 0816.11019

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