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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
quadruple with the property
is a set of four different non-zero elements of
that the product of any two distinct elements of this set
increased by
In the first chapter of this dissertation we considered the
existence of a
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
for integers
Also, there appeared some cases where sets cannot contain elements
of small norm, so it was necessary to consider the coefficients of
made those cases harder to handle. However, we obtained some artial results involving some congruence conditions modulo
During our examination, there appeared three sporadic possible exceptions i.e.
Therefore, in the second part of the dissertation it was reasonable to
consider the problem of the existence of
Using some known results about the extendibility of some families of
over the imaginary quadratic fields. Actually, if
following statements:\newline
{\bf i)} In
{\bf ii)} If
{\bf iii)} If
the form
{\bf iv)} If
For
By considering the extendibility of a
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: 1.186 *