An Investigation of Lucas Sequences

Publication Date

5-2007

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Major Advisor

Marion Scheepers

Comments

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)

(1.1)

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

sin(nθ + mθ) = cos nθ sin mθ + sin nθ cos mθ.

That the sequence {Vn, Un)} behaves like {cos nθ, sin nθ)} is very surprising. In fact, we will see that the sequence {cos nθ, sin nθ)} satisfies a recurrence similar to (1.1). Lucas sequences also have a curious connection to the solutions of the Pell equation x2 – Dy2 = 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) : x2 – Dy2 ≡m4}, 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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