Related papers: Very small intervals containing at least three pri…

Recently, several bounds have been obtained 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. Here,…

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We…

The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are…

The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in…

For a prime $p$, let $Z(p)$ be the smallest positive integer $n$ so that $p$ divides $F_{n}$, the $n$th term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for $\zeta(m)$, the density of primes $p$ for…

Let $M$ be a fixed positive integer. Let $(R_{j}(n))_{n\ge 1}$ be a linear recurrence sequence for every $j=0,1,\ldots, M$, and we set $f(n)=(R_0\circ \cdots \circ R_M)(n)$, where $(S\circ T)(n)= S(T(n))$. In this paper, we obtain…

We posit that $d_n^2 < 2p_{n+1}$ holds for all $n\geq 1$, where $p_n$ represents the $n$th prime and $d_n$ stands for the $n$th prime gap i.e. $d_n := p_{n+1} - p_n$. Then, the presence of a prime between successive perfect squares, as well…

Let $m(n, k)$ be the maximum number of prime implicants that any $k$-CNF on n variables can have. We show that $3^{n/3} \le m(n,2) \le (1+o(1))3^{n/3}$.

Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of McNew, Pollack and Pomerance, we establish the exact order of growth of $N(x,y)$ for all…

Let $\mathcal{P}$ be the set of primes and $\mathbb{N}$ the set of positive integers. Let also $r_1,...,r_t$ be positive real numbers and $R_2(r_1,...,r_t)$ the set of odd integers which can be represented as $$ p+2^{\lfloor…

We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must…

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can…

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty subsets of A which are relatively prime to n is \Phi(A, n) and the number of such…

Let $\PP^d$ be the $d$-fold direct product of the set of primes. We prove that if $A$ is a subset of $\PP^d$ of positive relative upper density then $A$ contains infinitely many "corners", that is sets of the form $\{x,x+te_1,...,x+te_d\}$…

We show that there is an infinite set of primes $\mathcal{P}$ of density one, such that the family of \textit{all} Cayley graphs of $\mathrm{SL}(2,p)$%, $p\in \mathcal{P}$, is a family of expanders.

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a…

We develop a simple $O((\log n)^2)$ test as an extension of Proth's test for the primality for $p2^n+1$, $p>2^n$. This allows for the determination of large, non-Sierpinski primes $p$ and the smallest $n$ such that $p2^n+1$ is prime. If $p$…

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we…

A primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either…

A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition (A,B) (called a split) with |A|, |B| >= 2 such that the set of edges joining A and B induces a complete bipartite graph. We prove that…