A primality criterion based on a Lucas' congruence

Let $p$ be a prime. In 1878 \'{E}. Lucas proved that the congruence $${p-1\choose k}\equiv (-1)^k\pmod{p}$$ holds for any nonnegative integer $k\in\{0,1,\ldots,p-1\}$. The converse statement was given in Problem 1494 of {\it Mathematics Magazine} proposed in 1997 by E. Deutsch and I.M. Gessel. In this note we generalize this converse assertion by the following result: If $n>1$ and $q>1$ are integers such that $${n-1\choose k}\equiv (-1)^k \pmod{q}$$ for every integer $k\in\{0,1,\ldots, n-1\}$, then $q$ is a prime and $n$ is a power of $q$.Comment: 6 page

Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$1+\frac 12 +\frac 13+...+\frac{1}{p-1}$$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2}$$ written in reduced form is divisible by $p$. 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

On a Convex Operator for Finite Sets

Let $S$ be a finite set with $n$ elements in a real linear space. Let \cJ_S be a set of $n$ intervals in \nR. We introduce a convex operator \co(S,\cJ_S) which generalizes the familiar concepts of the convex hull \conv S and the affine hull \aff S of $S$. We establish basic properties of this operator. It is proved that each homothet of \conv S that is contained in \aff S can be obtained using this operator. A variety of convex subsets of \aff S can also be obtained. For example, this operator assigns a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For \cJ_S which consists of bounded intervals, we give the upper bound for the number of vertices of the polytope \co(S,\cJ_S).Comment: 20 pages, 16 figure

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 ${n\choose m}$ modulo a prime $p$ in terms of the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the $p$-adic expansions of nonnegative integers $n$ and $m$, 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 $u$-nomial coefficients, the Gaussian $q$-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

Maximal and inextensible polynomials and the geometry of the spectra of normal operators

We consider the set S(n,0) of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ we let $|p|_{0}$ denote the distance from the origin to the zero set of $p'$. We determine all 0-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are inextensible, they are not necessarily locally maximal for Sendov's conjecture. This invalidates the recently claimed proofs of the conjectures of Sendov and Smale and shows that the method used in these proofs can only lead to (already known) partial results. In the second part of the paper we obtain a characterization of the critical points of a complex polynomial by means of multivariate majorization relations. We also propose an operator theoretical approach to Sendov's conjecture, which we formulate in terms of the spectral variation of a normal operator and its compression to the orthogonal complement of a trace vector. Using a theorem of Gauss-Lucas type for normal operators, we relate the problem of locating the critical points of complex polynomials to the more general problem of describing the relationships between the spectra of normal matrices and the spectra of their principal submatrices.Comment: A condensed version of the first half of this paper appeared in Math. Scand., see arXiv:math/0601600. Parts of the second half appeared in Trans. Amer. Math. Soc., see arXiv:math/0601519. The current version contains the full details of the counterexample constructions and some other result