Related papers: On Kurepa's problems in number Theory

For any prime $p$ and $\varepsilon>0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\varepsilon) \sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s)$ are…

Let $p$ be a large prime number. We prove that any integer $\lambda$ modulo $p$ can be represented in the form $$ m!n! +\sum_{i=1}^{47}n_i!\equiv \lambda \pmod p, $$ with $\max\{m,n,n_1,\ldots,n_{47}\}\ll p^{1300/1301}.$ This improves the…

Let $p$ be an odd prime and $r\geq 1$. Suppose that $\alpha$ is a $p$-adic integer with $\alpha\equiv2a\pmod p$ for some $1\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that…

Expressions of type $(p^q-1)/(p-1)$ and $a^2+ab+b^2$, where $a, b$ are natural and $p, q$ are prime numbers, are studied.

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[\zeta_m+\zeta_m^{-1}])$ and $K_{2k}(\mathbb{Z}[\zeta_m])$,…

We consider the $k$-error linear complexity of binary sequences derived from Eluer quotients modulo $2p$ ($p>3$ is an odd prime), recently introduced by J. Zhang and C. Zhao. We adopt certain decimal sequences to determine the values of…

In this paper, we solve a problem of Terence Tao. We prove that for any $K\geq 2$ and sufficiently large $N$, the number of primes $p$ between $N$ and $(1+\frac{1}{K})N$ such that $\mid kp+ja^{i}+l\mid$ is composite for all $1\leq a, |j|,…

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that…

We solve a theoretical arithmetics problem stated by Wac{\l}aw Sierpi\'nski. The problem has remained open for a couple of decades.

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…

We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p =…

This article examines the nontrivial solutions of the congruence \[ (p-1)\cdots(p-r) \equiv -1 \pmod p. \] We discuss heuristics for the proportion of primes $p$ that have exactly $N$ solutions to this congruence. We supply numerical…

In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation $n!=x^2-1$. It is conjectured that this equation has only three solutions, but up to now this is an open problem. Overholt observed…

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their $b$-adic expansion for some large positive integer $b$. The proof is based on the circle method, which relies on the Fourier structure of the integers with…

The classical notion of extreme $L_p$ discrepancy is a quantitative measure for the irregularity of distribution of finite point sets in the $d$-dimensinal unit cube. In this paper we find a dual integration problem whose worst-case error…

In a paper published by this author in www.academia.edu(see reference[3]), it was established that there exist no three positive integers which are consecutive terms of an arithmetic progression; and whose sum of squares is a perfect or…

Studying expressions of the form $(f(x)D)^p$, where $D={\displaystyle \frac{d}{dx}}$ is the derivative operator, goes back to Scherk's Ph.D. thesis in 1823. We show that this can be extended as ${\displaystyle\sum \gamma_{p;a}…

In this paper we show that the number of radial positive solutions of the following critical problem $$ \Delta_p u(x) + \lambda K(|x|) \,u(x) \, |u(x)|^{q-2} =0\,,$$ $$ u(x)>0 \quad |x|<1,$$ $$ u(x)=0 \quad |x|=1,$$ where $q=…

We give an historical account, including recent progress, on some problems of Erd\H os in number theory.

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ is a sufficiently small constant. Let $k(p,\,n)$ denote the least positive integer $k$ such that for…