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On the sum of consecutive cubes being a perfect square. (English) Zbl 0837.11012
In this paper, the author describes a systematic method that answers the following problem: Given $$n \in \mathbb{N}$$, 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.
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, J. Gebel, A. Pethö and 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 S. David for linear forms in elliptic logarithms [Mém. Soc. Math. Fr., Nouv. Sér. 62, 143 p. (1995; Zbl 0859.11048)]].
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 successfully 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.

MSC:
 11D25 Cubic and quartic Diophantine equations 11J86 Linear forms in logarithms; Baker’s method 11D85 Representation problems
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References:
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