arXiv subjects
Publications and source records attributed to Lionel Ponton.
Given a rational number $r$ such that $2r$ is not an integer, we prove that $\tan^2(rπ)$ is irrational unless it is equal to $0$, $1$, $3$ or $\frac{1}{3}$, using only basic trigonometry and the Rational Root Theorem. Moreover, we deduce that $\tan(rπ)$, $\cos^2(rπ)$ ans $\cos(rπ)$ are irrational numbers except in usual cases.
By using purely algebraic tools, we establish well-known properties of roots of Chebyshev polynomials. Especially, we show that these zeros are simple and lie in $(-1,1)$ and we prove in two ways that they are mostly irrational.
By using the Calkin-Wilf tree, we prove the irrationality of numbers of the form $α=\frac{\sqrt{N}+p}{q}$ where $N$ is a positive integer which is not a perfect square, $p$ is a rational integer such that $p^2<N$ and $q$ is a positive integer which divides $N-p^2$. For this, we consider an analogue of the Calkin-Wilf tree with root $α$ and we define a special path in this tree which satisfies remarkable properties of periodicity and symmetry. This path is closely related to the continued fraction expansion of $α$ and allows us to give new proofs of theorems due to Legendre and to Galois about the form of such an expansion in special cases of square roots and reduced quadratic surds.
We are interested in irrationality of roots for seven important families of polynomials : Tchebichef polynomials, Legendre polynomials, Laguerre polynomials, Hermite polynomials, Bessel polynomials, Bernoulli polynomials and Euler polynomials. One first proves, for the most part of them, existence of real roots then one studies the irrationality of them. The methods used are based on elementary arithmetical properties of algebraic numbers, some of them becoming from more general proofs that have allowed to derive the irreducibility of some of these polynomials.
On propose une méthode de résolution effective de l'équation diophantienne $(E_2): ax^2-by^2=1$ où $a$ et $b$ sont des entiers naturels non nuls et premiers entre eux. Cette méthode s'appuie sur le développement en fraction continuée de certains nombres irrationnels quadratiques que l'on décrit complètement. On commence par utiliser ces développements pour résoudre certaines équations de Pell-Fermat généralisées avant d'appliquer à l'équation $(E_2)$ les résultats obtenus.
We give two trees allowing to represent all positive rational numbers. These trees can be seen as ternary and quinary analogues of the Calkin-Wilf tree. For each of these two trees, we give recurrence formulas allowing to compute the rational number corresponding to the node n. These are analogues of the formulas given by Donald Knuth and Moshe Newman for the Calkin-Wilf tree. Finally, we show that the two sequences we have obtained, together with Calkin-Wilf sequence, are the only ones which satisfy a relation analogue to Newman's relation and enumerate the positive rationals.