The Diophantine equation \(x^{2p}-Dy^2=1\) and the Fermat quotient \(Q_p(m)\) (Q2742113)

Some results on the Diophantine equation (*) \(x^{2p}-Dy^2=1\), \(D\) a square-free positive integer, \(p\) a prime, are proved: (1) If \(D=q\) is a prime, then the only positive integers satisfying (*) are \((p,q,x,y)=(3,7,2,3),(2,5,3,4),(2,29,99,1820)\). (2) If \(D=2q\) (\(q\) a prime), then the only positive integers satisfying (*) are \((p,q,x,y)=(2,3,7,20),(5,61,3,22)\). NEWLINENEWLINENEWLINEThe proofs are based on lemma of Cao [Northeast Math. J. 2, No. 2, 219-227 (1986; Zbl 0669.10034)]\ concerning the integral solutions of \(x^p\pm 1=Dy^2\) and on a result of Ljunggren on the Diophantine equation \((x^p-1)/(x-1)=y^2\). The authors mainly refer only to \textit{Z. Cao}'s book [Introduction to Diophantine equations (Chinese), Haerbin Gongye Daxue Chubanshe, Harbin, (1989; Zbl 0849.11029)].NEWLINENEWLINENEWLINESeveral results are also stated without proof: (3) Equation (*) has no positive integer solution if \(p>2\) and either \(D=2qr\), \(q\), \(r\) odd primes, \(q\equiv r\equiv 5\bmod 8\), or \(D\) is not divisible by primes \(2mp+1\).NEWLINENEWLINENEWLINE(4) If \(p>3\), then the only positive integer solutions of the equation \(Q_p(m)=x^2\) are \((p,m,x)=(5,3,4),(7,2,3)\), while \(Q_p(m)=2x^2\) is impossible where \(Q_p(m)\) denotes the Fermat quotient \((m^{p-1}-1)/p\).