Let $\psi : \mathbb {N} \to [0,1/2]$ be given. The Duffin–Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|\alpha - p/q| < \psi (q)/q$ , provided series $\sum _{q=1}^\infty \varphi (q) / q$ is divergent. In present paper, we establish a quantitative version...

The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties. 1...

We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of congruent number curve. give upper bounds in each coset $2\mathcal{E}_D(\mathbb{Q})$ $\mathcal{E}_D(\mathbb{Q})$ and show that their total is $\ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})}$. further average non-torsion this family bounded above by $2$. As an application we also deduce from our system simultan...

Abstract Suppose s and t are coprime positive integers, let $$\sigma $$ σ be an -core partition $$\tau τ a partition. In this paper, we consider the set $${\mathcal {P}}_{\sigma ,\tau }(n)$$ P , (...

Let $$p<p'$$ be a pair of coprime positive integers. In this note, generalizing Morton’s work in the case $$\mathfrak {sl}_2$$ , we give formula for {sl}_r$$ Jones invariants torus knots $$T (p,p')$$ coloured with finite-dimensional irreducible representations $$L_r(n\Lambda _1)$$ . When $$r \le p$$ show that appropriate limits shifted (non-normalized, framing dependent) calculated along $$L_r(...

It is shown that Dirichlet's theorem on primes in an arithmetic progression is equivalent to the statement that every unit of a certain quotient ring Z of the nonstandard integers is the image of an infinite prime. The ring Z is the completion of Z relative to the "natural" topology on Z. 1. Notation. Throughout this note A shall denote the natural numbers, Z the rational integers, and P the po...

Let G be a finite group. A coprime commutator in is any element that can written as [x,y] for suitable x,y∈G such π(x)∩π(y)=∅. Here π(g) denotes the set of prime divisors order g∈G. An anti-coprime an [x,y], where π(x)=π(y). The main results paper are follows. If |xG|≤n whenever x commutator, then has nilpotent subgroup n-bounded index. every x∈G, H nilpotency class at most 4 [G:H] and |γ4(H)| ...

Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely coprime corings (coprime comodules). Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate their relatio...