On Pseudoprimes Related to Generalized Lucas Sequences

In this paper we consider the general sequences U„ and Vn satisfying the recurrences Un+2=mUn+l + Un, Vn+2=mV„+l+V„, (1.1) where m is a given positive integer, and UQ = 0, Ux = 1, V0 = 2, V1 = m. We shall occasionally refer to these sequences as U(m) and V(m) to emphasize their dependence on the parameter m. They can be represented by the Binet forms Un = {a-ni{a-P\ Vn = a+f3\ (1.2) where a+j3 = mmd aj3=-l, and we define A = 8 = (a-fl) = (a + /3)-4aj3 = m +4. When m = 1, these sequences reduce to F„ and Ln, with A = 5. Using (1.2), we can derive the identities (1.3) through (1.7), which correspond to wellknown formulas that are proved, for instance, in [11]: U2„=U„V„, (1.3) V2n = K-2(-iy, (1.4) AU„=V„-4(-i)\ (1.5) 2Un+s = U„Vs+V„Us, (1.6) 2Vn+s = V„Vs + AU„Us. (1.7) When «is a prime/?, we have Vp = a+P = (a+P) = m (mod/?), and using Fermat's "little theorem," this gives the well-known result Vp = m (mod p), when p is prime. (1.8) Any composite numbers n satisfying the corresponding equation Vn=m(moAri) (1.9) are called pseudoprimes. Di Porto and Filipponi [7] have called such numbers Fibonacci Pseudoprimes of the m* kind (m-F.Psps), whereas Bruckman [2] has called them Lucas Pseudoprimes. As a compromise, we shall call them V(m)-pseudoprimes or V(m)-psp. In the case of m= 1, when V„ becomes Ln, it has been proved that all V(l)-psp's are odd: see {14], [5], and [3 J. In the more general case, since the interest in V(/w)-psps relates to tests for primality, only odd V(/w)psp's will be considered, as in [7], and we shall restrict the definition of pseudoprimes to odd composite n satisfying (1.9).