Squares in Lehmer sequences and some Diophantine applications (Q2759144)

Let \(L>0\) and \(M\) be rational integers such that \(L-4M> 0\) and \(\gcd (L,M)=1\). Let \(\alpha\) and \(\beta\) be the roots of \(x^2- \sqrt{L}x+m\). For a nonnegative integer \(n\), define NEWLINE\[NEWLINEP_n= P_n(\alpha,\beta)= \begin{cases} \frac{\alpha^n-\beta^n} {\alpha-\beta} &\text{for \(n\) odd},\\ \frac{a^n-\beta^n} {\alpha^2-\beta^2} &\text{for \(n\) even}. \end{cases}NEWLINE\]NEWLINE The sequence \(\{P_n\}\) is known as the Lehmer sequence who studied this extensively [{\textit{D. H. Lehmer}, Ann. Math. (2) 31, 419-448 (1930; JFM 56.0874.04)]. In this paper the authors investigate the occurrence of squares and certain square classes in Lehmer sequences. \textit{A. Rotkiewicz} [Acta Arith. 42, 163-187 (1983; Zbl 0519.10004)] studied the equation NEWLINE\[NEWLINEP_p= px^2, \qquad p\text{ odd prime}. \tag{1}NEWLINE\]NEWLINE He showed that (1) has no solution if either \((L,M)\equiv (1,0)\pmod 4\) and \((\frac{M}{L})= 1\) or if \((L,M)\equiv (0,3)\pmod 4\) and \((\frac{L}{M})= 1\). Further he showed that NEWLINE\[NEWLINEP_p= x^2, \qquad p\text{ odd prime}\tag{2}NEWLINE\]NEWLINE has no solution if \((L,M)\equiv (3,0)\pmod 4\) and \((\frac{M}{L})= 1\) or if \((L,M)\equiv (0,1)\pmod 4\) and \((\frac{L}{M})= 1\). NEWLINENEWLINENEWLINEIn the present work, the authors use the results of Rotkiewicz on Jacobi symbols to show that NEWLINENEWLINENEWLINE(1) has no solution if \((L,M)\equiv (2,1)\pmod 4\) and \((\frac{L}{M})= 1\) and NEWLINENEWLINENEWLINE(2) has no solution if \((L,M)\equiv (2,1)\pmod 4\) and \((\frac{L}{M})= 1\) if \(p>3\). NEWLINENEWLINENEWLINEFurther they apply their result to refine a result of Ljunggren on the Diophantine equation \(mX^2- nY^4= 2\). They also solve completely the equations NEWLINE\[NEWLINE(X^2+1) (Y^2+1)= Z^4, \quad (X^2+1) (Y^2-1)= Z^4, \quad(X^2-1) (Y^2-1)= Z^4.NEWLINE\]NEWLINE}