Related papers: Exponential sums over integers without large prime…

Using combinatorial techniques, we derive a recurrence identity that expresses an exponential power sum with negative powers in terms of another exponential power sum with positive powers. Consequently, we derive a formula for the power sum…

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same…

We obtain a new bound in the uniform version of the Glasner property for matrices with polynomial entries, improving that of K. Bulinski and A. Fish (2021). This improvement is based on a more careful examination of complete rational…

Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is…

We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a…

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+\delta}$ which have conveniently sized divisors. The main feature of…

Let $u_{2j}$ be the product of the odd positive integers $< 2j$. For $n$ an integer $\ge 1$, define \[ f(x)=\sum_{j=0}^{n}a_j\frac{x^{2j}}{u_{2j+2}}, \] where the $a_j$'s are arbitrary integers with $|a_0|=1$. In 1929, I. Schur established…

The famous irreducibility criteria of Sch\"onemann-Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper we provide several irreducibility criteria of…

The best deterministic unconditionally proven integer factorization algorithms have exponential running time complexities of O(N^(1/4)) arithmetic operations, and conditional on the Riemann hypothesis, there is a deterministic algorithm of…

We obtain upper bounds on the number of solutions to congruences of the type $$ (x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. We give some applications…

It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer $N=(4k+1)^{4m+1}\prod_{i=1}^\ell ~ q_i^{2\alpha_i}$ to establish that there do…

A set of integers greater than 1 is primitive if no member in the set divides another. Erd\H{o}s proved in 1935 that the series $f(A) = \sum_{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets $A$. In 1986 he asked…

Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to…

We prove a vanishing theorem for the p-adic cohomology of exponential sums on affine space. In particular, we obtain new classes of exponential sums on affine space that have a single nonvanishing p-adic cohomology group. The dimension of…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…

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…

For an element $a$ of an integral domain D under an equivalence relation \tau, the \tau-factorization of a is defined as \lambda a_1 a_2... a_k, where \lambda is a unit in D and a_i \tau a_j for all i, j. An irreducible element has no…

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian…