On the Diophantine equation \(x (x + 1) \dots (x + n) + 1 = y^2\). (English) Zbl 0996.11022

Abe, Nobuhisa

Proc. Japan Acad., Ser. A 76, No. 2, 16-17 (2000).

The author determines all the integer solutions of the title equation for \(1\leq n\leq 15\), \(n\) odd. His method eventually relies on Runge’s method (see e.g. the paper of A. Grytczuk and A. Schinzel [Colloq. Math. Soc. János Bolyai 60, 329-356 (1992; Zbl 0849.11033)] and the references given there, or the paper of P. G. Walsh [Acta Arith. 62, 157-172 (1992; Zbl 0769.11017)] for a quantitative variant of the method).
For \(n\) odd, the title equation is a special case of \[ f(x)= g(y) \quad\text{in }x,y\in \mathbb{Z}, \tag{1} \] where \(f,g\in \mathbb{Z}[x]\) are monic polynomials, and \(\deg(g)\mid \deg(f)\). The method of solution of (1) was completely described by L. Szalay [Superelliptic equations of the form \(y^p= x^{kp}+ a_{kp-1} x^{kp-1}+\cdots+ a_0\), Bull. Greek Math. Soc. (to appear)] and the case when \(\deg(g)= 2\), earlier by L. Szalay [Acta Acad. Paedagog. Agriensis, Sect. Mat. 27, 19-24 (2000; Zbl 0973.11039)]. The actual way of treatment in the latter paper is very similar to the one followed by the author.
None of the above or other relevant references are cited in the article; the author only refers to the well-known paper of P. Erdős and J. L. Selfridge [Ill. J. Math. 19, 292-301 (1975; Zbl 0295.10017)].

Reviewer: L.Hajdu (Debrecen)

MSC:

11D41

Higher degree equations; Fermat’s equation

Keywords:

consecutive integers; Runge’s method

Citations:

Zbl 0849.11034; Zbl 0849.11033; Zbl 0769.11017; Zbl 0973.11039; Zbl 0295.10017

References:

[1]	Erdös, P. and Selfridge, J. L.: The Product of consecutive integers is never a power. Illinois J. Math., 19 , 292-301 (1975). · Zbl 0295.10017