On generalized Lucas pseudoprimes (Q1820806)

Let \(R=R(A,B)=\{R_ n\}^{\infty}_{n=0}\) be a Lucas sequence of integers defined by \(R_ n=A R_{n-1}-B R_{n-2}\) \((n>1)\), where A and B are integers and the initial terms are \(R_ 0=0\), \(R_ 1=1\). We say that R(A,B) is non-degenerate if \(D=A^ 2-4B\neq 0\) and \(R_ n(A,B)\neq 0\) for all \(n>0\). For given non-zero integer D and for any positive integers n, k with \((n,2D)=1\) we denote n-k(D/n) by \(\delta _ k(n)\), where (D/n) is the Jacobi symbol; furthermore let \(n^ *=n/\prod _{p| n}p.\) The following results are proved. (I) For any given non-zero integer D and positive integers n, k with \((n,2D)=1\) and \(n\geq k\) there exist \((n^ *,k)\prod _{p| n}\{(\delta _ k(n),\delta _ 1(n))-1\}\) distinct values of A (mod n), for which there is an integer B such that \(A^ 2-4B\equiv D\) (mod n) and \((*)\quad R_{\delta _ k(n)}(A,B)\equiv 0\quad (mod n).\) (II) For any given non-zero integer D, any positive integer \(k\neq 2\) there exists infinitely many non-degenerate Lucas sequences R(A,B) such that the congruence (*) has an infinitude of composite solutions n.