Successioni di Lucas, Polinomi Ciclotomici e Self-Divisors

This thesis is a collection of some classic and new topics concerning Lucas sequences, with particular regard to the prime factorization of their terms. Accordingly with the title, the thesis consists of three chapters, entitled respectively ``Lucas sequences'', ``Cyclotomic polynomials'', and ``Self-divisors of Lucas sequences''. In the first chapter, we introduce Lucas sequences $(u_n)_{n \geq 0}$ as special linearly recurrent sequences of order $2$ satisfying $u_n = a u_{n-1} + b u_{n-2}$, $n \geq 2$, for some relative prime integers $a$ and $b$, with the initial conditions $u_0 = 0$ and $u_1 = 1$. We show that, for our purposes, there is no loss of generality in assuming that $(u_n)_{n \geq 0}$ is ``nondegenerate'', i.e., the ratio of the two roots $\alpha$ and $\beta$ of the characteristic polynomial $X^2 - aX - b$ is not a root of unity. We prove the basic divisibility properties of Lucas sequences, especially that they are ``strong divisibility sequence'', i.e., $\gcd(u_m, u_n) = u_{\gcd(m,n)}$, for all positive integers $m$ and $n$. Furthermore, we show that for each positive integer $m$ it is well-defined the ``rank of apparition'' of $m$ in $(u_n)_{n \geq 0}$, denoted by $\tau(m)$, that is the least positive integer $n$ such that $m \mid u_n$. Finally, we give a detailed study of the $p$-adic valuation $\upsilon_p(u_n)$ of the general term $u_n$ of our Lucas sequence, showing how it depends on $\upsilon_p(u_{\tau(p)})$ and $\upsilon_p(n)$. In the second chapter, we define the $n$-th cyclotomic polynomial $\Phi_n(X)$ and its homogenized version $\Phi_n(X, Y)$, showing their basic properties. In particular, we prove that $\Phi_n(X)$ has integer coefficients and is irreducible over the rationals. Then we illustrate the connection between cyclotomic polynomials and Lucas sequences given by the factorization formula $$u_n = \prod_{\substack{d \mid n \\ d > 1}} \Phi_d \quad \text{for } n \geq 1,$$ where $\Phi_d := \Phi_d(\alpha, \beta)$ are called ``Sylvester cyclotomic numbers'' and are proven to be integers. A study of the several properties of the prime factors of Sylvester cyclotomic numbers is performed. In particular, we prove that they are ``almost'' pairwise relatively primes. Precisely, if $\gcd(\Phi_m, \Phi_n)>1$ for some integers $m > n \geq 2$, then $\gcd(\Phi_m, \Phi_n)$ is a prime number. Tangentially, we show how Sylvester cyclotomic numbers can be used to prove two special cases of Dirichlet's theorem on primes in arithmetic progressions, namely, that for each integer $n \geq 2$ there exist infinitely many prime numbers $\equiv 1$ (mod $n$) (respectively, $\equiv -1$ (mod $n$)). Finally, for real $\alpha$ and $\beta$, we prove the result known as Primitive Divisor Theorem, which states that for each $n > 12$ the term $u_n$ has a primitive divisor, i.e., a prime factor $p$ such that $p \nmid u_m$ for any positive integer $m<n$. The last chapter is the main original contribution of the thesis. In it we deal with ``self-divisors'' of Lucas sequences, that is, positive integers $n$ such that $n \mid u_n$. This topic has already been considered by several authors, e.g., André-Jeannin, Somer, Alba González, Luca, Pomerance, and Shparlinski. We prove that for any fixed Lucas sequence the number of self-divisors $\leq x$ is less than $$x^{1-\left(\frac1{2}+o(1)\right)\frac{\log\log\log x}{\log \log x}},$$ as $x \to +\infty$. This generalizes a previous result of Luca and Tron about the self-divisors of Fibonacci numbers.