It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t\left(x\right)=x^4-t{x}^3-6x^2+tx+1$, assuming that $t>0$, $t\ne 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal...

This is an exposition of the recent work of Bugeaud, Hanrot and Mihăilescu showing that Catalan’s conjecture can be proved without using logarithmic forms and electronic computations.

The subject of the talk is the recent work of Mihăilescu, who proved that the equation $x^p-y^q=1$ has no solutions in non-zero integers $x,y$ and odd primes $p,q$. Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to $x^u-y^v=1$ in integers $x,y\>0$ and $u,v\>1$ is $3^2-2^3=1$. Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...

We shall describe how to construct a fundamental solution for the Pell equation $x^2-m{y}^2=1$ over finite fields of characteristic $p\ne 2$. Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation $x^2-m{y}^2=n$.

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.This paper contains proofs of conjectures made in [16] on class number 2 and what this author has dubbed the Euler-Rabinowitsch polynomial for real quadratic fields. As well, we complete the list of Richaud-Degert types given in [16] and show how the behaviour of the Euler-Rabinowitsch polynomials and certain continued fraction expansions come into play in the complete determination...

The purpose of this paper is to prove that the common terms of linear recurrences $M(2a,-1,0,b)$ and $N(2c,-1,0,d)$ have at most $2$ common terms if $p=2$, and have at most three common terms if $p>2$ where $D$ and $p$ are fixed positive integers and $p$ is a prime, such that neither $D$ nor $D+p$ is perfect square, further $a,b,c,d$ are nonzero integers satisfying the equations $a^2-D{b}^2=1$ and $c^2-(D+p)d^2=1$.

The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \mathbb{N}$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $min\{x,y,z\}>1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation$$\begin{array}{cc}\hfill {\Phi }_n(x,y)& =x^3+(n^8+2n^6-3n^5+3n^4-4n^3+5n^2-3n+3)x^{2}y\hfill \\ & -(n^3-2)n^{2}x{y}^2-y^3=\pm 1,\hfill \end{array}$$for $n\ge 0$.