37,737 research outputs found
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
Variations of Kurepa's left factorial hypothesis
Kurepa's hypothesis asserts that for each integer the greatest
common divisor of and is . Motivated by an
equivalent formulation of this hypothesis involving derangement numbers, here
we give a formulation of Kurepa's hypothesis in terms of divisibility of any
Kurepa's determinant of order
by a prime . In the previous version of this article we have proposed
the strong Kurepa's hypothesis involving a general Kurepa's determinant
with any integer . We prove the ``even part'' of this hypothesis which
can be considered as a generalization of Kurepa's hypothesis. However, by using
a congruence for involving the derangement number with an odd
integer , we find that the integer is a
counterexample to the ``odd composite part'' of strong Kurepa's hypothesis.
We also present some remarks, divisibility properties and computational
results closely related to the questions on Kurepa's hypothesis involving
derangement numbers and Bell numbers.Comment: 18 pages. This is the previous (first) version of the article
extended with Section 4 where we disprove the "odd composite part''of Strong
Kurepa's hypothesi
Harmful and toxic algae
The chapter provides basic facts about harmful and toxic algae. It also discusses the conditions that stimulate their occurrence, different types of harmful and toxic algal blooms and their effects to fish and marine environment. The different strategies in coping with the problem of harmful and toxic algal blooms are also discussed
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