Related papers: Compositions with 3 Pairwise Coprime Parts

Let $\{ {U_{n}\}_{n \geq 0} }$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\ldots, p_s\}$ be fixed prime numbers and $\{b_1,\ldots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain…

We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes…

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation…

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with…

We show that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\leqslant m$, the expression $$ \frac{1}{[n_1]}{n_1+n_{m}\brack n_1}^{-1}…

We study the properties of the product, which runs over the primes, $$\mathfrak{p}_n = \prod_{s_p(n) \, \geq \, p} p \quad (n \geq 1),$$ where $s_p(n)$ denotes the sum of the base-$p$ digits of $n$. One important property is the fact that…

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let…

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\,n_2,\,n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split…

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 $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations…

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with $x,y,z$ positive integers. The \emph{Erd\H{o}s-Straus conjecture} asserts that…

A k-composition of n is a sequence of length k of positive integers summing up to n. In this paper, we investigate the number of k-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic…

We show that for any irrational $\alpha$ and any $\tau<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|n\alpha\|\leq n^{-\tau}.$$ We also show that for all sufficiently large $b$ there exist 3-digit…

A linear combination $aT_r(m)+bT_s(n)$ of an \mbox{$r$-gonal} number $T_r(m)$ and an $s$-gonal number $T_s(n)$ with mutually coprime positive integer coefficients $a$ and $b$ produces infinitely many primes as $m$ and~$n$ varies over the…

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4, 6$ and each choice of the signs our equation has infinitely many co-prime positive…

Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique.…

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^\alpha C^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for "most" primes $q$ and suitably large prime…

Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all…

We refine a result of W.P. Li and Wang on the values of the form $ \lambda_1p_1 + \lambda_2p_2^{2} + \lambda_3p_3^{2} + \mu_1 2^{m_1} +...+ \mu_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers,…