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...

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 $n\ge 0$.

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 $a$.