Power values of products of consecutive integers and binomial coefficients (Q2707564)

In this article the author gives a description of the latest developments concerning power values of products of consecutive integers, binomial coefficients and related questions. After a brief historical introduction the remarkable result of Erdős and Selfridge is stated. This result says that for any integer \(k\geq 2\) a product of \(k\) consecutive integers is never a power; i.e., the equation NEWLINENEWLINE(1) \(n(n+1)\cdots(n+k-1)=x^l\) in integers \(n\geq 1, k,x,l\geq2\), has no solution. A related equation treated by Erdős is NEWLINENEWLINE(2) \(\binom{n+k-1}{k} =x^l\) in integers \(k\geq 2, n\geq k+1, x,l\geq2\). NEWLINENEWLINEIn 1951 Erdős showed that for \(k\geq 4\) equation (2) has no solution. The method of Erdős does not work for equation (2) if \(k<4\). The author shows how the case \(k=2\) follows from a result of \textit{H. Darmon} and \textit{L. Merel} [J. Reine Angew. Math. 490, 81--100 (1997; Zbl 0976.11017)] and also settles the case \(k=3\). The equations (1) and (2) can be combined as NEWLINENEWLINE(3) \(n(n+1)\cdots(n+k-1)=bx^l\) in integers \(n,b,\;x\geq1, k,l\geq2\), with \(P(b)\leq k,\) where \(b\) is an \(l\)th power, and \(P(b)\) denotes the greatest prime factor of \(b.\) NEWLINENEWLINEBy a result of the reviewer for \(k\geq 4\), and Györy for \(k\leq3\), we have that equation (3) has only the solution \((n,k,b,x, l)=(48,3,6,140,2)\) with \(P(x)>k\) apart from the case \(k=b= l=2.\) Another generalisation of equation (3) is NEWLINENEWLINE(4) \(n(n+d)\cdots(n+(k-1)d)=bx^l\) in integers \(n,b,\;x\geq1, k\geq3, l\geq2\), with \({\gcd}(n,d)=1, P(b)\leq k\). NEWLINENEWLINEA conjecture of Erdős says that equation (4) implies that \(k\) is bounded by an absolute constant. In fact, this constant is believed to be 3. This equation has been studied by several authors. The results are partial. The reviewer has shown that equation (4) with \( l \geq 3\) and \(d \leq 6\) has no solution. By the works of Saradha, Hajdu and Filakovszky, it is known that equation (4) with \(l=2, b=1\) and \(d\leq30\) has as its only solution \((n,d,k,x)=(18,7,3,120)\) and \((1,24,3,35)\). The author shows in this article that equation (4) with \(b=1, k=3\) and \(l\geq3\) has no solution by using the results of Darmon, Merel and Ribet.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].