Related papers: An Idempotent Cryptarithm

Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second…

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…

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\geq 2$, $m\neq 4$, there are…

For each integer $b \geq 3$ and every $x \geq 1$, let $\mathcal{N}_{b,0}(x)$ be the set of positive integers $n \leq x$ which are divisible by the product of their nonzero base $b$ digits. We prove bounds of the form $x^{\rho_{b,0} + o(1)}…

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…

Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial Dirichlet character modulo $m$, there are…

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 $\sqrt{n}$.

In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$…

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case where…

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…

For an integer $b\geq 2$, a positive integer is called a $b$-Niven number if it is a multiple of the sum of the digits in its base-$b$ representation. In this article, we show that every arithmetic progression contains infinitely many…

Let b > 1 be an integer and denote by s_b(m) the sum of the digits of the positive integer m when is written in base b. We prove that s_b(n!) > C_b log n log log log n for each integer n > e, where C_b is a positive constant depending only…

Let b $\ge$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\le$ N : |s 3 (n) -- s 2 (n)| $\le$ 0.1457205 log n} \textgreater{} N…

From known effective bounds on the prime counting function of the form \[ |\pi(x)-\mathrm{Li}(x)| < a \;x \;(\ln x)^{b} \; \exp\left(-{c}\; \sqrt{\ln x}\right); \qquad (x \geq x_0); \] it is possible to establish exponentially tight…

In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties…

We investigate the ratio $\rho_{n,L}$ of prefix codes to all uniquely decodable codes over an $n$-letter alphabet and with length distribution $L$. For any integers $n\geq 2$ and $m\geq 1$, we construct a lower bound and an upper bound for…

We exhibit a new application of two dimensional covering systems, examples of integer pairs $a,b$ for which $a^m-b^n$ has a prime divisor from some given finite set of primes, for every pair of integers $m,n\geq 0$. This leads us to…

For $x\geq 3$, we define $w(x)$ as the highest integer $w$ for which there exist integers $m, y\geq 1$ and $1\leq n_1<\dots<n_m\leq x$ such that $n_1\cdots n_m=y^w$. We show that \[w(x)=x\exp\big(-(\sqrt{2}+o(1))\sqrt{\log x\log\log…

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that…

Conceptually, a rough number is a positive integer with no small prime factors. Formally, for real numbers $x$ and $y$, let $\Phi(x,y)$ denote the number of positive integers at most $x$ with no prime factors less than $y$. In this paper we…