Almost perfect powers in products of consecutive integers
A first main object of the present paper is the study of the equation \[ m(m+1)\cdots (m+k-1)=by^n \tag{1} \] in the integer unknowns \(m,b,y\geq 1\) and \(n\geq 2\), where \(k\geq 2\) is an integer parameter. Since the publication, in 1975, of a famous result by Erdős and Selfridge, which says that the product of consecutive positive integers is never a perfect power, a rather extensive research around this and similar equations has been done and a number of very interesting relevant papers has been published. Let us denote by \(P(x)\) the greatest prime divisor of the integer \(x\) and by \(p_k\) the \(k\)-th prime. The first main result of the paper (Theorem 1) is the following: Under the restrictions \(P(b)\leq p_k\) and (only for the case \(k=2\)) \(n>2\), equation (1) can have solutions only in the following cases: (i) \(k=2\) and \(m\in\{1,2,3,8\}\), (ii) \(k=3\) and \(m\in\{1,2,3,4,8\}\), (iii) \(k=4\) and \(m\in\{1,\ldots, 7\}\), (iv) \(k=5\) and \(m\in\{1,\ldots, 8\}\). A second main object of the paper is the study of the equation (below the reviewer somewhat modifies the authors' notation) \[ m(m+d_1)\cdots (m+d_{k-2})=by^n\tag{2,} \] where \(k\geq 3\), in the posititve integer unknowns \(m,b\), \(d_1<\cdots <d_{k-2}\) and \(n\geq 2\) (with \(n>2\) when \(k=3\)). If \(k\in\{3,4,5\}\), then Theorem 2 gives explicitly in a table all cases for which equation (2) is possible under the restrictions \(d_{k-2}\leq k-1\) and \(P(b)\leq p_{k-1}\). For example, when \(k=4\), equation (2) is possible only with \((d_1,d_2)=(1,3)\) and \(m\in\{1,2,3,5,9,15,24\}\). For the proof of Theorem 2 the authors need to prove Theorem 3 which reads as follows: Let \(n\) be equal to 4 or to an odd prime. Then the equation \(m(m+3)=2^{\alpha}3^{\beta}y^n\) in integers \(m,y\geq 1\) and \(\alpha,\beta\geq 0\) is possible only for \(m=1,3,6,9,24\), except possibly if \(n\in I=\{19,23,29,31,37,41,43,47,73\}\) in which case further values of \(m\), with \(m\equiv 1,5\pmod{6}\) and \(\alpha=1,\beta=0\), cannot be excluded (but no such solutions are known). Theorem 3 is, in turn, based on the following Theorem 4 which the authors also prove: Let \(n\) be equal to 4 or to an odd prime not belonging to the set \(I\) (see above). Then, solutions \((x,y)\) to the Diophantine equation \(x^n-2y^n=\pm 3\), distinct from \(\pm (-1,1)\), exist only when \(n=3\), in which case there is one further solution, namely \((x,y)=\pm (5,4)\). For the proof of Theorem 4 a crucial role is played by a sharp lower bound for homogeneous linear forms in two logarithms of algebraic numbers, due to M.~Laurent, M.~Mignotte and Y.~Nesterenko, as well as a lower bound of M.~Bennett for the distance of \((a/b)^{1/n}\) from the nearest rational number (under certain restrictions on the integers \(a,b\) and \(n\)). For the proof of Theorem 3 several lemmas from the literature are used. It is worth noticing, for example, that the authors use the following result: If \(n\geq 7\) is a prime number and \(\alpha\) is an integer \(\geq 2\) then the equation \(x^n+2^{\alpha}y^n=3z^2\) has no solution in non-zero coprime integers \(x,y,z\). This is contained in a more general recent (2004) result by \textit{M. A. Bennett} and \textit{C. Skinner} [Can. J. Math. 56, No. 1, 23--54 (2004; Zbl 1053.11025)], for the proof of which Galois representations and modular forms are used. For the proofs of Theorems 1 and 2 the authors again combine various existing results, the most impressive of which being, probably, \textit{J. M.~Gandhi} 's [J. Reine Angew. Math. 256, 163--167 (1972; Zbl 0248.10018)] sufficient condition for non-existence of non-zero solutions to equations \(x^n+y^n=cz^n\) with \(n\) prime and \(n|z\). The corrections concern the replacement of Lemma 6.
scientific article; zbMATH DE number 2175898