Lucas sequences with cyclotomic root field (Q2846556)

Let \(P\) and \(Q\) be integers (\(Q\neq 0\)). The classical Lucas sequences \(X=(U_n)_{n\geq 0}\) and \(X=(V_n)_{n\geq 0}\), respectively, satisfy the recurrence NEWLINE\[NEWLINEX_{n+2}=PX_{n+1}-QX_n,\qquad n\geq 0NEWLINE\]NEWLINE with initial values \((U_0, U_1)=(0,1)\) and \((V_0,V_1)=(2,P)\). Much is known about identities, arithmetic properties and applications for these two sequences. The paper under review is devoted to the study of sequences that arise for special values of the discriminant \(D=P^2-4Q\), namely, when \(D=-E^2\), or \(D=-3F^2\) for integers \(E\) and \(F\). The author defines six sequences (\(X=G,H, S, T, Y, Z\)), all of them satisfying the recurrence relation above, and shows that many of the properties of the classical Lucas sequences can be carried over to these sequences. The initial values for these sequences are given by NEWLINE\[NEWLINEX_0=1, \qquad X_1=(P\pm E)/2, \;\; (P\pm F)/2, \;\; (P\pm 3F)/2, \text{ respectively}.NEWLINE\]NEWLINENEWLINENEWLINEThe paper is divided into five chapters where each chapter is devoted to a special aspect of the theory. The sequences are defined in Chapter 2 where various elementary identities are given. Chapter 3 deals with arithmetic properties such as congruence formulas (for instance, \(X_p \pmod p\)), laws of appearance and repetition, and powers of 2 and 3 in \(X\). In Chapter 4, the author gives Wolstenholme type congruences for the six sequences under consideration. Chapter 5 is devoted to the study of the set of indices \(n\) such that \(n\) divides \(X_n\). In the final chapter (Chapter 6), results on prime densities are given. The prime density of \(X\) is the limit (relative within the set of primes), if it exists, of the number of primes that divide some term of \(X\).