Possible Restricted Periods of Certain Lucas Sequences Modulo P

Let (u)=u(a,b), called the Lucas sequence of the first kind (LSFK), be the second-order linear recurrence satisfying 1 $$ {u_{n + 2}} = a{u_{n + 1}} + b{u_{{n^,}}} $$ where uo = 0,u 1 = 1, and the parameters aandb are integers. Let (v)=v (a,b),called the Lucas sequence of the second kind (LSSK), be the recurrence satisfying (1) with the initial termsv o = 2, v 1 = a. Let 2 $${x^2} - ax - b$$ be the characteristic polynomial of u (a, b) and v(a, b) with characteristic roots r 1 and r 2. Let D =a 2 + 4b = (r 1– r 2)2 be the discriminant of u(a, b) and v (a, b). By the Binet formulas, 3 $${u_n} = \frac{{r_1^n - r_2^n}}{{{r_1} - {r_2}}} = \frac{{r_1^n - r_2^n}}{{ \pm \sqrt D }} $$ if D≠0 and 4 $${v_n} = r_1^n + r_{{2^.}}^n $$ Throughout this paper, p will denote an odd prime and P will denote a prime ideal of $$Q(\sqrt D ) $$ dividing p.