230 research outputs found
-adic quotient sets
For , the question of when is dense in the positive real numbers has been examined by
many authors over the years. In contrast, the -adic setting is largely
unexplored. We investigate conditions under which is dense in the
-adic numbers. Techniques from elementary, algebraic, and analytic number
theory are employed in this endeavor. We also pose many open questions that
should be of general interest.Comment: 24 page
On the p-adic denseness of the quotient set of a polynomial image
The quotient set, or ratio set, of a set of integers is defined as . We consider the case in which
is the image of under a polynomial ,
and we give some conditions under which is dense in .
Then, we apply these results to determine when is dense in
, where is the set of numbers of the form , with integers. This allows us to answer a
question posed in [Garcia et al., -adic quotient sets, Acta Arith. 179,
163-184]. We end leaving an open question
Power Values of Certain Quadratic Polynomials
In this article we compute the th power values of the quadratic
polynomials with negative squarefree discriminant such that is coprime
to the class number of the splitting field of over . The theory
of unique factorisation and that of primitive divisors of integer sequences is
used to deduce a bound on the values of which is small enough to allow the
remaining cases to be easily checked. The results are used to determine all
perfect power terms of certain polynomially generated integer sequences,
including the Sylvester sequence.Comment: 16 Pages; corrected and expanded versio
Approximating cube roots of integers, after Heron's Metrica III.20
Heron, in Metrica III.20-22, is concerned with the the division of solid
figures - pyramids, cones and frustra of cones - to which end there is a need
to extract cube roots. We report here on some of our findings on the conjecture
by Taisbak in C.M.Taisbak, Cube roots of integers. A conjecture about Heron's
method in Metrika III.20. Historia Mathematica, 41 (2014), 103-104
Symmetries of Monocoronal Tilings
The vertex corona of a vertex of some tiling is the vertex together with the
adjacent tiles. A tiling where all vertex coronae are congruent is called
monocoronal. We provide a classification of monocoronal tilings in the
Euclidean plane and derive a list of all possible symmetry groups of
monocoronal tilings. In particular, any monocoronal tiling with respect to
direct congruence is crystallographic, whereas any monocoronal tiling with
respect to congruence (reflections allowed) is either crystallographic or it
has a one-dimensional translation group. Furthermore, bounds on the number of
the dimensions of the translation group of monocoronal tilings in higher
dimensional Euclidean space are obtained.Comment: 26 pages, 66 figure
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