Related papers: Counting carefree couples

Let 0<\theta<\pi such that \cos\theta\in \Q. In this paper, we prove that for given positive square-free coprime integers k,l, there exist infinitely many pairs (M,N) of \theta-congruent numbers such that lN=kM. This generalize the previous…

A set A of integers is said to be sum-free if there are no solutions to the equation x + y = z with x,y and z all in A. Answering a question of Cameron and Erdos, we show that the number of sum-free subsets of {1,...,N} is O(2^(N/2)).

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) =…

It is not difficult to find an asymptotic formula for the number of pairs of positive integers $x, y \le H$ such that $x^2 + y^2 + 1$ is squarefree. In the present paper we improve the estimate for the error term in this formula using the…

The (classical) crosscorrelation is an important measure of pseudorandomness of two binary sequences for applications in communications. The arithmetic crosscorrelation is another figure of merit introduced by Goresky and Klapper…

We prove that every integer greater than two may be written as the sum of a prime and a square-free number.

A king invites n couples to sit around a round table with 2n+1 seats. For each couple, the king decides a prescribed distance d between 1 and n which the two spouses have to be seated from each other (distance d means that they are…

For a given base $g\ge2$, a positive integer is called a palindrome if its base $g$ expansion reads the same backwards as forwards. In this paper, we give an asymptotic formula for the number of relatively prime pairs of palindromes of a…

We study the random Fibonacci tree, which is an infinite binary tree with non-negative integers at each node. The root consists of the number 1 with a single child, also the number 1. We define the tree recursively in the following way: if…

We measure whether there are numerous pairs of twin primes (hereafter referred to as twin prime pairs) according to the prime number inferred by sieve of Eratosthenes. In this study, we reveal at least three additional twin prime pairs…

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…

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs…

This is a continuation of the study, begun by Ceccherini-Silberstein and Woess, of context-free pairs of groups and the related context-free graphs in the sense of Muller and Schupp. Instead of the cones (connected components with respect…

Translated from the Latin original, "De numeris amicabilibus" (1747). E100 in the Enestroem index. Euler starts by saying that with the success of mathematical analysis, number theory has been neglected. He argues that number theory is…

Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$…

We consider the class of Berge graphs that contain no odd prism and no square (cycle on four vertices). We prove that every graph G in this class either is a clique or has an even pair, as conjectured by Everett and Reed. This result is…

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

In this paper we introduce a simple method of searching for the prime pairs in the famous Goldbach Conjecture. The method, which is based on certain integer identities as well as an observation related to the remainder property, enables us…

Let A be a pre-defined set of rational numbers. We say a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of…