Vijest
NEW DOCTORAL DEGRESS Some diophantine problems over the imaginary quadratic fields
Ivan Soldo
orcid.org/0000-0002-6195-6626
; Department of Mathematica, University of osijek, Osijek, Croatia
Sažetak
Let $z$ be an element of a commutative ring $R$. A Diophantine
quadruple with the property $D(z)$, or a $D(z)$-quadruple,
is a set of four different non-zero elements of $R$ with the property
that the product of any two distinct elements of this set
increased by $z$ is a square of some element in $R$.
In the first chapter of this dissertation we considered the
existence of a $D(z)$-quadruple in the ring $\mathbb{Z}[\sqrt{-2}]$.
We tried to extend previous results on this subject by Abu Muriefah
and Al-Rashed, from 2004. We obtained several new polynomial formulas
for Diophantine quadruples with the property $D(a+b\sqrt{-2}\,)$,
for integers $a$ and $b$ satisfying certain congruence conditions.
Also, there appeared some cases where sets cannot contain elements
of small norm, so it was necessary to consider the coefficients of
$z$ concerning modulus greater than the usual one. Thus, this
made those cases harder to handle. However, we obtained some artial results involving some congruence conditions modulo $11$ on $a$ and $b$.
During our examination, there appeared three sporadic possible exceptions i.e. $z \in \{-1, 1\pm 2\sqrt{-2}\}$. Note that $1\pm 2\sqrt{-2}= -1 \cdot (1 \mp \sqrt{-2})^2$, so the existence of $D(-1)$-quadruples would imply the existence of $D(1+2\sqrt{-2})$ and
$D(1-2\sqrt{-2})$-quadruples.
Therefore, in the second part of the dissertation it was reasonable to
consider the problem of the existence of $D(-1)$-quadruples in $\mathbb{Z}[\sqrt{-2}\,]$ and in the ring of integers in other quadratic fields.
Using some known results about the extendibility of some families of
$D(-1)$-pairs over integers, we obtained similar results
over the imaginary quadratic fields. Actually, if $t>1$, we proved the
following statements:\newline
{\bf i)} In $\mathbb{Z}[\sqrt{-t}]$, a $D(-1)$-quadruple of the form
$\{1,2,c,d\}$ does not exist.\newline
{\bf ii)} If $b\in\{5,10,26,50\}$ and $t \ne b-1$, in $\mathbb{Z}[\sqrt{-t}]$
$D(-1)$-quadruple of the form $\{1,b,c,d\}$ does not exist.\newline
{\bf iii)} If $t \not \in \{4,16\}$, in $\mathbb{Z}[\sqrt{-t}]$ a $D(-1)$-quadruple of
the form $\{1,17,c,d\}$ does not exist.\newline
{\bf iv)} If $t \not \in \{4,9,36\}$, in $\mathbb{Z}[\sqrt{-t}]$ a $D(-1)$-quadruple of the form $\{1,37,c,d\}$ does not exist.
For $t=1$ and other exceptions of (ii), (iii) and (iv), we also proved that there exist infinitely many $D(-1)$-quadruples of the form $\{1,b,-c,d\}$, $c,d>0$ in $\mathbb{Z}[\sqrt{-t}]$.
By considering the extendibility of a $D(-1)$-pair $\{1,17\}$ in
$\mathbb{Z}[\sqrt{-2}]$ and a $D(-1)$-pair $\{1,37\}$ in $Z[\sqrt{-3}]$, we observed that there are three possibilities for positivity of elements $c$ and $d$. This led us to form the system of simultaneous Pellian equations and our attempt to find all solutions over the integers. In that way,
we used results from simultaneous diophantine approximations, linear forms in
logarithm of algebraic numbers and Baker-Davenport reduction.
Ključne riječi
Hrčak ID:
93304
URI
Datum izdavanja:
5.12.2012.
Posjeta: 907 *