Almost perfect powers in arithmetic progression (Q2759139)

Let \(n\), \(d\), \(k\), \(t\), be positive integers such that \(\gcd(n,d)= 1\), \(k\geq 2\), \(t\geq 2\), and \(r= k-t\in \{0,1\}\). For any integer \(\nu>1\), let \(P(\nu)\) be the greatest prime factor of \(\nu\). Further, let \(d_1,\dots, d_t\) be integers with \(0\leq d_1<\cdots< d_t< k\). Also put \(\Delta= (n+d_1d)\dots (n+d_td)\). First, for \(r=1\), \(d>1\) and \(k\geq 4\), the authors give all tuples \((n,d,k,d_1,\dots, d_t)\) for which \(P(\Delta)\leq k\). Next the authors consider the equation (1) \(\Delta=by^\ell\) in positive integers \(b,d,k,\ell,n,y,d_1,\dots, d_t\) with \(\ell> 2\) prime, \(P(b)\leq k\), \(P(\Delta)> k\), and subject to the restrictions given above. NEWLINENEWLINENEWLINEA classic result about (1) is that of \textit{P. Erdős} and \textit{J. L. Selfridge} [Ill. J. Math. 19, 292-301 (1975; Zbl 0295.10017)], who showed that for \(r=0\), \(b=d=1\) this equation has no solutions. Their result was extended for \(b\geq 1\), \(k\geq 4\) by \textit{N. Saradha} [Acta Arith. 82, 147-172 (1997; Zbl 0922.11025)] and for \(k=2,3\) by \textit{K. Győry} [Acta Arith. 83, 87-92 (1998; Zbl 0896.11012)]. In the present paper, the authors completely solve equation (1) for \(r=d=b=1\). Another result is that, given (1) with \(k\geq 4\) if \(r=0\), and \(k\geq 0\) if \(r=1\), the maximal divisor \(D_1\) of \(d\) such that all prime divisors of \(D_1\) are \(1\pmod\ell\) is greater than 1. Further refinements of existing results are obtained, but their formulations are too involved to reproduce here.