Related papers: On primes represented by quartic polynomials on av…
We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the…
This paper introduces a novel class of prime-generating quadratic polynomials defined by $f_{Z,k,H}(n) = n^2 - (2Zk - 1)n + \frac{(2Zk - 1)^2 + H}{4}$, where $Zk \in \mathbb{Z}_{\geq 0}$ and $H$ belongs to the set of Heegner numbers. This…
This paper presents algorithms for calculating the quadratic character and the norms of prime ideals in the ring of integers of any quadratic field. The norms of prime ideals are obtained by means of a sieve algorithm using the quadratic…
We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…
We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{\alpha_1}\cdots p_s^{\alpha_s} q_1^{\beta_1}\cdots q_t^{\beta_t}$ with $\beta_1, \ldots, \beta_t$ bounded by a given constant.…
We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…
We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The…
Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of…
The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…
We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a…
We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes.
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the…
The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.
We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent…
We present a novel conjecture concerning the additive representation of natural numbers using prime powers. Based on extensive computational verification, we conjecture that every integer n > 23 can be expressed as a sum of at most five…
Gerard and Washington proved that, for $k > -1$, the number of primes less than $x^{k+1}$ can be well approximated by summing the $k$-th powers of all primes up to $x$. We extend this result to primes in arithmetic progressions: we prove…
We calculate bounds for orthant probabilities for the equicorrelated multivariate normal distribution and use these bounds to show the following: for degree $k>4$, the probability that a $k$-homogeneous polynomial in $n$ variables attains a…
We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.
Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with…