An Investigation of Lucas Sequences

Publication Date


Type of Culminating Activity


Degree Title

Master of Science in Mathematics



Major Advisor

Marion Scheepers


In 1878 Édouard Lucas published a paper which summarized much of his research into the theory of what he called “simply periodic numerical functions.” These were pairs of integer sequences which exhibit many properties characteristic of the trigonometric functions, and which he put to use in developing a new type of primality test. One way to define these “Lucas sequences” is to pick some integers P and Q, and put: (V0, U0) = (2, 0), (V1, U1) = (P, 1), and (for n > 1)


(Vn+1, Un+1) = P(Vn, Un) - Q(Vn-1, Un-1)

(here addition and scalar multiplication are performed component-wise). From these definitions one can show (and we will see) that

(where Δ = P2 — 4Q). If we put , these become

Compare this last pair of identities with the trigonometric identities

cos( + ) = cos cos + (-1) sin sin ,

sin( + ) = cos sin + sin cos .

That the sequence {Vn, Un)} behaves like {cos , sin )} is very surprising. In fact, we will see that the sequence {cos , sin )} satisfies a recurrence similar to (1.1). Lucas sequences also have a curious connection to the solutions of the Pell equation x2Dy2 = 1 (actually this is a generalization of the connection to the trig functions): when Q = 1 (recall that {Un} and {Vn} are defined in terms of P and Q), we have

for all n; and in groups of the form {(x, y) : x2Dy2m4}, every cyclic subgroup can be described in terms of {Un} and {Vn}. Another interesting fact regarding these sequences is that {Un} shares many of the same divisibility properties as the sequence of integers. For example Um|Un iff m|n; moreover gcd(Um, Un) = (Ugcd(m,n). Combining this last fact with the fact that almost every prime q (that is, all odd primes not dividing P2 – 4Q) divides either Uq-1 or Uq+1, we have the following:

Theorem. Let m > 1 be odd and ε = ±1. If m|Um-ε but for each prime q dividing m – ε, then m is prime.

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