Related papers: Divisibility of binomial coefficients and generati…
By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several…
Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…
We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.
In this letter, we prove an inequality involving alternating binomial logarithmic sums by exploiting the variance of the logarithm of the maximum of independent and identically distributed exponential random variables. This inequality was…
We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct…
We consider an extremal problem for polynomials, which is dual to the well-known Smale mean value problem. We give a rough estimate depending only on the degree.
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $\Phi^*_n(q)$, which is closely related to the cyclotomic polynomial $\Phi_n(x)$ and to primitive prime…
We investigate the distribution of the function $\omega(n)$, the number of distinct prime divisors of $n$, in residue classes modulo $q$ for natural numbers $q$ greater than 2. In particular we ask `prime number races' style questions, as…
Ideals generated by adjacent 2-minors are studied. First, the problem when such an ideal is a prime ideal as well as the problem when such an ideal possesses a quadratic Gr\"obner basis is solved. Second, we describe explicitly a primary…
We complete a proof of a theorem that was inspired by an Indian Olympiad problem, which gives an interesting characterization of a prime number $p$ with respect to the binomial coefficients ${n\choose p}$. We also derive a related result…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results…
Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…
We study and describe possibilities for arities of elementary theories and of their expansions. Links for arities with respect to Boolean algebras, to disjoint unions and to compositions of structures are shown. The dynamics for arities of…
In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.
We introduce the $B$-Stirling numbers of the first and second kind, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as…
We give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary polynomials.
We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary…
We present a new, elementary, dynamical proof of the prime number theorem.