Related papers: Prime-bounded subwords

This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…

Let p be any prime, and $p^(\nu_p(n!))$ the maximal power of $p$ dividing $n!$. It is proved that there exists a positive integer $n_0$, which depends only on $p$, such that $q^(\nu_q(n!)) < p^(\nu_p(n!))$ for all $n \ge n_0$ and all primes…

Recent results of Bourgain and Shparlinski imply that for almost all primes $p$ there is a multiple $mp$ that can be written in binary as $mp= 1+2^{m_1}+ \cdots +2^{m_k}, \quad 1\leq m_1 < \cdots < m_k,$ with $k=66$ or $k=16$, respectively.…

We prove that every $n$-letter word over $k$-letter alphabet contains some word as a subsequence in at least $k^{n/4k(1+o(1))}$ many ways, and that this is sharp as $k\to\infty$. For fixed $k$, we show that the analogous number deviates…

The main result of this thesis is to show that there are only finitely many integers $n$ such that both $n$ and $d(n)$ are highly composite numbers at the same time, where $d(n)$ is the divisor function. Bertrand's postulate [4] is used…

Let P denote the set of all primes. Suppose that P_1, P_2, P_3 are three subsets of P with the sum of their lower densities relative to P is greater than 2. We prove that for sufficiently large odd integer n, there exist p_i\in P_i such…

In this paper, it is proved that every sufficiently large even integer can be represented as the sum of two squares of primes, two cubes of primes, two biquadrates of primes and 16 powers of 2. Furthermore, there are at least 5.313% odd…

For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 0} \} $ and let $s(a,b)=ab-a-b$. It is well known that all integers which are greater than $s(a,b)$ are in $T(a,b)$. Let $\pi (a, b)$ be the number of…

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby…

Let $C(n)$ denote the number of permutations $\sigma$ of $[n]=\{1,2,\dots,n\}$ such that $\gcd(j,\sigma(j))=1$ for each $j\in[n]$. We prove that for $n$ sufficiently large, $n!/3.73^n < C(n) < n!/2.5^n$.

In the present work we investigate the largest possible gaps between consecutive numbers which can be written as the difference of two primes. The best known upper bounds are the same as those concerning the largest possible difference of…

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many…

This note is an attempt to attack a conjecture of Fraenkel and Simpson stated in 1998 concerning the number of distinct squares in a finite word. By counting the number of (right-)special factors, we give an upper bound of the number of…

An positive even number is said to be a Kronecker number if it can be written in infinitely many ways as the difference between two primes, and it is believed that all even numbers are Kronecker numbers. We will study the division and…

Let $p_1 = 2, p_2 = 3,...$ be the sequence of all primes. Let $\epsilon$ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers $q \ge 3$ and $a$. We will establish a lower bound for the number of primes…

A Toda prime of an integer $n$ is an odd prime $p$ such that $4n=(p-1)k$ with $k$ coprime to $p$. We conjecture that every positive integer admits at least two Toda primes. We give a partial proof that every positive integer admits at least…

We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.

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…

A non-empty word $w$ is a \emph{border} of a word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ is \emph{privileged} if $\vert u\vert\leq 1$ or if $u$ has a privileged border $w$ that appears…

We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we…