Related papers: Primes in Quadratic Progressions on Average

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval…

We consider weighted averages of the number of representations of an even integer as a sum of two prime numbers, where each summand lies in a given arithmetic progression modulo a common integer $q$. Our result is uniform in a suitable…

We prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes.

Let $f(x)$ be a square free quartic polynomial defined over a quadratic field $K$ such that its leading coefficient is a square. If the continued fraction expansion of $\displaystyle \sqrt{f(x)}$ is periodic, then its period $n$ lies in the…

This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…

This article describes the mean curvature flow, some of the discoveries that have been made about it, and some unresolved questions.

This is an earlier, but more general, version of "An L^1 Ergodic Theorem for Sparse Random Subsequences". We prove an L^1 ergodic theorem for averages defined by independent random selector variables, in a setting of general…

The article describes prime intervals into the prime factorization of the middle binomial coefficient. Prime factors and prime powers are distributed in layers. Each layer consists of non-repeated prime numbers which are chosen (not…

We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

Triangular distributions are a well-known class of distributions that are often used as an elementary example of a probability model. Maximum likelihood estimation of the mode parameter of the triangular distribution over the unit interval…

In this paper, we give a complete description of the associated primes of each power of the edge ideal of an increasing weighted tree.

In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted…

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems,…

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…

I present a new property of prime numbers that leads to a generalization of Cramer's conjecture. The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain…

Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…

For a fixed number field $K$, we consider the mean square error in estimating the number of primes with norm congruent to $a$ modulo $q$ by the Chebotar\"ev Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using a…

We apply recent bounds of the author (math.PR/0609224) for generalized Smirnov statistics to the distribution of integers whose prime factors satisfy certain systems of inequalities.