26 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
Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
In 1862 Wolstenholme proved that for any prime the numerator of the
fraction written in reduced form is divisible by , and the numerator of
the fraction
written in reduced form is divisible by . The first of the above
congruences, the so called {\it Wolstenholme's theorem}, is a fundamental
congruence in combinatorial number theory. In this article, consisting of 11
sections, we provide a historical survey of Wolstenholme's type congruences and
related problems. Namely, we present and compare several generalizations and
extensions of Wolstenholme's theorem obtained in the last hundred and fifty
years. In particular, we present more than 70 variations and generalizations of
this theorem including congruences for Wolstenholme primes. These congruences
are discussed here by 33 remarks.
The Bibliography of this article contains 106 references consisting of 13
textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of
Integer Sequences. In this article, some results of these references are cited
as generalizations of certain Wolstenholme's type congruences, but without the
expositions of related congruences. The total number of citations given here is
189.Comment: 31 pages. We provide a historical survey of Wolstenholme's type
congruences (1862-2012) including more than 70 related results and 106
references. This is in fact version 2 of the paper extended with congruences
(12) and (13
The congruence of Wolstenholme and generalized binomial coefficients related to Lucas sequences
Using generalized binomial coefficients with respect to fundamental Lucas
sequences we establish congruences that generalize the classical congruence of
Wolstenholme and other related stronger congruences.Comment: 23 page
Multivariate Ap\'ery numbers and supercongruences of rational functions
One of the many remarkable properties of the Ap\'ery numbers ,
introduced in Ap\'ery's proof of the irrationality of , is that they
satisfy the two-term supercongruences \begin{equation*}
A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes . Similar congruences are conjectured to hold for all Ap\'ery-like
sequences. We provide a fresh perspective on the supercongruences satisfied by
the Ap\'ery numbers by showing that they extend to all Taylor coefficients of the rational function \begin{equation*}
\frac{1}{(1 - x_1 - x_2) (1 - x_3 - x_4) - x_1 x_2 x_3 x_4} . \end{equation*}
The Ap\'ery numbers are the diagonal coefficients of this function, which is
simpler than previously known rational functions with this property.
Our main result offers analogous results for an infinite family of sequences,
indexed by partitions , which also includes the Franel and
Yang--Zudilin numbers as well as the Ap\'ery numbers corresponding to . Using the example of the Almkvist--Zudilin numbers, we further indicate
evidence of multivariate supercongruences for other Ap\'ery-like sequences.Comment: 19 page
Congruences concerning Jacobi polynomials and Ap\'ery-like formulae
Let be a prime. We prove congruences modulo for sums of the
general form and
with . We also consider the
special case of the former sum, where the congruences hold
modulo .Comment: to appear in Int. J. Number Theor
q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon
We establish a supercongruence conjectured by Almkvist and Zudilin, by
proving a corresponding -supercongruence. Similar -supercongruences are
established for binomial coefficients and the Ap\'{e}ry numbers, by means of a
general criterion involving higher derivatives at roots of unity. Our methods
lead us to discover new examples of the cyclic sieving phenomenon, involving
the -Lucas numbers.Comment: Incorporated comments from referees. Accepted for publication in Int.
J. Number Theor