Related papers: Counting carefree couples
Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman…
A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function $\pi_2(x)$, the number of semiprimes less than or equal to $x$, is studied. Using a combinatorial argument,…
Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci-…
We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[\alpha n]$, $[\alpha n]+1$, where $\alpha>1$ is irrational number with bounded partial quotient or irrational algebraic number.
Friezes patterns are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their…
We prove that for every simple theory $T$ (or even simple thick compact abstract theory) there is a (unique) compact abstract theory $T^\fP$ whose saturated models are the lovely pairs of $T$. Independence-theoretic results that were proved…
A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…
Let $m$ be a rational integer with $m \neq 0, \pm 1$, and consider the pure number field $K = \mathbb{Q}(\sqrt[n]{m})$ with $n \ge 3$. Most papers discussing the monogenity of pure number fields focus exclusively on the case where $m$ is…
Let $a$ and $b$ be positive integers with prime factorisations $a = p_1^np_2^n$ and $b = q_1^nq_2^n$. We prove that the number of essentially distinct $\alpha$-graceful labelings of the complete bipartite graph $K_{a, b}$ equals the…
Given a family $\mathcal{F}$ of subsets of $[n]$, we say two sets $A, B \in \mathcal{F}$ are comparable if $A \subset B$ or $B \subset A$. Sperner's celebrated theorem gives the size of the largest family without any comparable pairs. This…
A number $n$ is practical if every integer in $[1,n]$ can be expressed as a subset sum of the positive divisors of $n$. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and…
We relate the size of the error term in the Hardy-Littlewood conjectured formula for the number of prime pairs to the $L^{1}$ norm of an exponential sum over the primes formed with the von Mangoldt function.
The well-known expansion of rational integers in an arbitrary integer base different from $0, 1, -1$ is exploited to study relations between numerical monoids and certain subsemigroups of the multiplicative semigroup of nonzero integers.
We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all…
We consider the four structures $(\mathbb{Z}; \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Z}; <, \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Q}; \mathrm{Sqf}^\mathbb{Q})$, and $(\mathbb{Q}; <, \mathrm{Sqf}^\mathbb{Q})$ where $\mathbb{Z}$ is the additive…
Goldbach`s Conjecture, "every even number greater than 2 can be expressed as the sum of two primes" is renamed Goldbach`s Rule for it can not be otherwise. The conjecture is proven by showing that the existence of prime pairs adding to any…
We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. We prove that the composition of two cofree coalgebras is again cofree, and we give sufficient…
The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were…
Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$ we show the existence of a relation between the sets of the first elements of the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for every…