#### Title

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

#### Recommended Citation

Hinkel, Dustin E., "An Investigation of Lucas Sequences" (2007). *Boise State University Theses and Dissertations*. 567.

https://scholarworks.boisestate.edu/td/567

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## 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

PandQ, and put: (V_{0},U_{0}) = (2, 0), (V_{1},U_{1}) = (P, 1), and (forn> 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 =

P^{2}— 4Q).If we put , these becomeCompare this last pair of identities with the trigonometric identities

cos(

nθ+mθ) = cosnθcosmθ+ (-1) sinnθsinmθ,sin(

nθ+mθ) = cosnθsinmθ+ sinnθcosmθ.That the sequence {

V,_{n}U)} behaves like {cos_{n}nθ, sinnθ)} is very surprising. In fact, we will see that the sequence {cosnθ, sinnθ)} satisfies a recurrence similar to (1.1). Lucas sequences also have a curious connection to the solutions of the Pell equationx^{2}–Dy^{2}= 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 ofPandQ), we havefor all

n; and in groups of the form {(x,y) :x^{2}–Dy^{2}≡m4}, every cyclic subgroup can be described in terms of {U} and {_{n}V}. Another interesting fact regarding these sequences is that {_{n}U} shares many of the same divisibility properties as the sequence of integers. For example_{n}Uiff_{m}|U_{n}m|n; moreover gcd(U,_{m}U) = (_{n}U_{gcd(m,n)}. Combining this last fact with the fact that almost every primeq(that is, all odd primes not dividingP^{2}– 4Q) divides eitherU_{q}_{-1}orU_{q}_{+1}, we have the following:Theorem.Letm> 1 be odd and ε = ±1. Ifm|Ubut for each prime_{m-ε}qdividingm –ε, thenmis prime.