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 $n\ge 0$.

Let $M$ be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields $K$ with mixed signature having power integral bases and containing $M$ as a subfield. We also determine all generators of power integral bases in $K$. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for $M=\mathbb{Q}\left(\sqrt{2}\right),\mathbb{Q}\left(\sqrt{3}\right),\mathbb{Q}\left(\sqrt{5}\right).$

It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.

This paper investigates the system of equations $$x^2+a{y}^m=z_1^2,x^2-a{y}^m=z_2^2$$ in positive integers $x$, $y$, $z_1$, $z_2$, where $a$ and $m$ are positive integers with $m\ge 3$. In case of $m=2$ we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient $a$ with the condition $gcd(x,z_1)=gcd(x,z_2)=gcd(z_1,z_2)=1$. Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero $a$.