Related papers: Primes in Sumsets

In the number $373$ all subwords ($3$, $7$, $37$, $73$, and $373$) are prime. Similarly, in $9719$ all subwords are divisible by at most one prime. And similarly again in $7319797913$ all subwords are divisible by at most two primes. These…

Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb{F}_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb{F}_p})^n$ be an arbitrary subset such that $$\{…

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (\alpha…

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 1/2, then every sufficiently large even integer can be written as the sum of eight primes from A. The constant 1/2 in this statement is…

Pilz's conjecture states that for any finite set $A=\{a_1,a_2,\dots,a_k\}$ of positive integers and positive integer $n$ in the union of the sets $\{a_1,2a_1,\dots,na_1\},\dots, \{a_k,2a_k,\dots,na_k\}$ (considered as a multiset) at least…

The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a…

We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite…

We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.

Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. The proof is based on a bespoke container…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n = (a_k a_{k-1} \cdots a_0)_b$ in base $b$ with $a_k \neq 0$ has the symmetric property $a_{k-i} = a_i$ for every…

We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\ge 5$ and $r\ge 2$, we prove that \begin{equation} \sum\limits_{\begin{smallmatrix}…

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each…

The Piatetski-Shapiro sequences are of the form $\mathcal{N}_{c} := (\lfloor n^{c} \rfloor)_{n=1}^\infty$, where $\lfloor \cdot \rfloor$ is the integer part. It is expected that there are infinitely many primes in a Piatetski-Shapiro…

Let $A$ be a finite multiset of integers. If $B$ be a multiset such that $A$ and $B$ are $t$-complementing multisets of integers, then $B$ is periodic. We obtain the Biro-type upper bound for the smallest such period of $B$: Let…

We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then \[|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}\] for some absolute constant $c>0$. This improves upon a result of…

Let $A\subset [1,x]$ be a non-empty set of primes with $|A|= \alpha x(\log x)^{-1}$. We prove that there exist absolute constants $c_1,c_2>0$ such that, as $x$ gets sufficiently large, we have $|A+A|\geq c_1(\log x)(\log \log…

Let $h$ be a positive integer and $A, B_1, B_2,\dots, B_h$ be finite sets in a commutative group. We bound $|A+B_1+...+B_h|$ from above in terms of $|A|, |A+B_1|,\dots,|A+B_h|$ and $h$. Extremal examples, which demonstrate that the bound is…