425 research outputs found
A universal sequence of integers generating balanced Steinhaus figures modulo an odd number
In this paper, we partially solve an open problem, due to J.C. Molluzzo in
1976, on the existence of balanced Steinhaus triangles modulo a positive
integer , that are Steinhaus triangles containing all the elements of
with the same multiplicity. For every odd number ,
we build an orbit in , by the linear cellular automaton
generating the Pascal triangle modulo , which contains infinitely many
balanced Steinhaus triangles. This orbit, in , is
obtained from an integer sequence called the universal sequence. We show that
there exist balanced Steinhaus triangles for at least of the admissible
sizes, in the case where is an odd prime power. Other balanced Steinhaus
figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal
trapezoids or lozenges, also appear in the orbit of the universal sequence
modulo odd. We prove the existence of balanced generalized Pascal triangles
for at least of the admissible sizes, in the case where is an odd
prime power, and the existence of balanced lozenges for all admissible sizes,
in the case where is a square-free odd number.Comment: 30 pages ; 10 figure
Balanced simplices
An additive cellular automaton is a linear map on the set of infinite
multidimensional arrays of elements in a finite cyclic group
. In this paper, we consider simplices appearing in the
orbits generated from arithmetic arrays by additive cellular automata. We prove
that they are a source of balanced simplices, that are simplices containing all
the elements of with the same multiplicity. For any
additive cellular automaton of dimension or higher, the existence of
infinitely many balanced simplices of appearing in
such orbits is shown, and this, for an infinite number of values . The
special case of the Pascal cellular automata, the cellular automata generating
the Pascal simplices, that are a generalization of the Pascal triangle into
arbitrary dimension, is studied in detail.Comment: 33 pages ; 11 figures ; 1 tabl
Prime arithmetic Teichmuller discs in H(2)
It is well-known that Teichmuller discs that pass through "integer points''
of the moduli space of abelian differentials are very special: they are closed
complex geodesics. However, the structure of these special Teichmuller discs is
mostly unexplored: their number, genus, area, cusps, etc. We prove that in
genus two all translation surfaces in H(2) tiled by a prime number n > 3 of
squares fall into exactly two Teichmuller discs, only one of them with elliptic
points, and that the genus of these discs has a cubic growth rate in n.Comment: Accepted for publication in Israel Journal of Mathematics. A previous
version circulated with the title "Square-tiled surfaces in H(2)''. Changes
from v1: improved redaction, fixed typos, added reference
Lucas' theorem: its generalizations, extensions and applications (1878--2014)
In 1878 \'E. Lucas proved a remarkable result which provides a simple way to
compute the binomial coefficient modulo a prime in terms of
the binomial coefficients of the base- digits of and : {\it If is
a prime, and are the
-adic expansions of nonnegative integers and , then
\begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}.
\end{equation*}}
The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of
Lucas}), plays an important role in Number Theory and Combinatorics. In this
article, consisting of six sections, we provide a historical survey of Lucas
type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas
like theorems for some generalized binomial coefficients, and some their
applications.
In Section 1 we present the fundamental congruences modulo a prime including
the famous Lucas' theorem. In Section 2 we mention several known proofs and
some consequences of Lucas' theorem. In Section 3 we present a number of
extensions and variations of Lucas' theorem modulo prime powers. In Section 4
we consider the notions of the Lucas property and the double Lucas property,
where we also present numerous integer sequences satisfying one of these
properties or a certain Lucas type congruence. In Section 5 we collect several
known Lucas type congruences for some generalized binomial coefficients. In
particular, this concerns the Fibonomial coefficients, the Lucas -nomial
coefficients, the Gaussian -nomial coefficients and their generalizations.
Finally, some applications of Lucas' theorem in Number Theory and Combinatorics
are given in Section 6.Comment: 51 pages; survey article on Lucas type congruences closely related to
Lucas' theore
On Quasi-Periodicity in Proth-Gilbreath Triangles
Let PG be the Proth-Gilbreath operator that transforms a sequence of integers
into the sequence of the absolute values of the differences between all pairs
of neighbor terms. Consider the infinite tables obtained by successive
iterations of PG applied to different initial sequences of integers. We study
these tables of higher order differences and characterize those that have
near-periodic features. As a biproduct, we also obtain two results on a class
of formal power series over the field with two elements F2 that can be
expressed as rational functions in several ways
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