Related papers: There are infinitely many (-1,1)-Carmichael number…

In the present paper we prove that for any fixed $1<c<7/6$ there exist infinitely many consecutive square-free numbers of the form $[n^c], [n^c]+1$ and we also establish an asymptotic formula in given interval.

For a Carmichael number $n$ with prime factors $p_1,\cdots,p_m$, define $$K=GCD[p_1-1,\cdots,p_m-1],$$ and let $C_\nu(X)$ denote the number of Carmichael numbers up to $X$ such that $K=\nu$. Assuming a strong conjecture on the first prime…

In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \leq k \leq n-1$, the binomial coefficient $\binom{n}{k}$ is…

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…

For every prime $p$, we construct an infinite countable group that contains precisely $p-1$ elements which are not $p$th powers.

In 1960, W. Sierpinski proved that there are infinitely many positive odd numbers $k$, such that for any positive integer $n$, $k\times2^n+1$ is a composite number. Such numbers are called "Sierpinski numbers". In this study, by using…

A composite positive integer $n$ is said to be a {\it weak Carmichael number} if $$ \sum_{\gcd(k,n)=1\atop 1\le k\le n-1}k^{n-1}\equiv \varphi(n) \pmod{n}. \leqno(1) $$ It is proved that a composite positive integer $n$ is a weak Carmichael…

In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given…

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. We give an easy and new proof of the fact that the set of primes $p$ such that $p \mid P(n)$, for some $n \in \mathbb{Z}$, is infinite. We also get analog of this result for some special…

We prove that for each odd number k, the sequence (k2^n+1)_{n\ge 1} contains only a finite number of Carmichael numbers. We also prove that k=27 is the smallest value for which such a sequence contains some Carmichael number.

Let $[\, \cdot\,]$ be the floor function. In this paper we show that when $1<c<\frac{3849}{3334}$, then there exist infinitely many prime numbers of the form $[n^c]$, where $n$ is square-free.

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…

We prove that there are infinitely many integers, which can represent as sum of a square-free integer and a prime $p$ with $||\alpha p+\beta||<p^{-1/10}$, where $\alpha$ is irrational.

The number of primes of a kind x^2+1 is infinite.

We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive function $\theta(c)$ with the property that the largest prime factor of $\fl{n^c}$…

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…

Suppose that $\alpha,\beta\in\mathbb{R}$. Let $\alpha\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which…

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZ-algebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine…

In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.

We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer $n$, which is not a prime power, is an elliptic Carmichael number for a random curve…