A second order linear recurrence is a sequence { a n } \{ {a_n}\} of integers satisfying a a n + 2 = M a n + 1 − N a n {a_{n + 2}} = M{a_{n + 1}} - N{a_n} where N and M are fixed integers and at least one a n {a_n} is nonzero. If k is an integer, then the number m ( k ) m(k) of solutions of a n = k {a_n} = k is at most 3 (respectively 4) if M 2 − 4 N > 0 {M^2} - 4N > 0 and there is an odd prime q ≠ 3 q \ne 3 (respectively q = 3) such that q | M q|M and q ∤ k N q\nmid kN . Further M = sup k integer m ( k ) M = {\sup _k}{\;_{{\text {integer}}}}m(k) is either infinite or ≤ 5 \leq 5 provided that either (i) ( M , N ) = 1 (M,N) = 1 or (ii) 6 ∤ N 6\nmid N .