In this paper, as a main theorem, we prove that the decision version of the Frobenius problem is Σ2 -complete under Karp reductions. Given a finite set A of coprime positive integers, we call the greatest integer that cannot be represented as a nonnegative integer combination of A the Frobenius number, and we denote it as g(A). We call a problem of finding g(A) for a given A the Frobenius probl...

The primary aim of this paper is to study the generalized Fermat equation \[ x^2+y^{2n} = z^{3p} \] in coprime integers $x$, $y$, and $z$, where $n \geq 2$ $p$ a fixed prime. Using modularity results over totally real fields explicit computation Hilbert cuspidal eigenforms, we provide complete resolution case $p=7$, obtain an asymptotic result for $p$. Additionally, using similar techniques, so...

We prove that the number of fractions $$h_1/h_2$$ integers $$h_1,h_2$$ a subset $$\mathcal {A}\subset \mathcal {H}\cap [1,X]$$ is at least $$ \alpha X/(\log X)^{3/2}$$ , where {H}$$ set $$p-1$$ p being prime such $$p+1$$ sum two coprime squares. So, this $$\gg _\varepsilon ^{1+\varepsilon }|\mathcal {A}|^2$$ $$\varepsilon any positive real number. take opportunity to describe geometrical view s...

We work on an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points \(F\) sum divisors function \(\sigma : {\mathbb{F}}_2[x] \mapsto {\mathbb{F}}_2[x]\) (defined mutatis mutandi like usual integers) form \(F := A^2 \cdot S\), \(S\) square-free, with \(\omega(S) \leq 3\), coprime \(A\), for \(A\) even, whatever degree, under some conditions. This ...

Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number of principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa|n which...

Abstract We describe a computationally efficient approach to resolving equations of the form $$C_1x^2 + C_2 = y^n$$ C 1 x 2 + = y n </...

In this paper we give an overview over recent developments (initiated by P. Corvaja and U. Zannier in [3]) on Diophantine problems where linear recurring sequences are involved and which were solved by using W.M. Schmidt’s Subspace Theorem. Moreover, as a new application, we show: let (Gn) and (Hn) be linear recurring sequences of integers defined by Gn = c1α n 1 + c2α n 2 + · · ·+ ctα t and Hn...

Adding new classes of integers to literature is both challenging and charming. Until a class completely characterized, mathematics never going be worth it. While it's absurd play with without intended consequences. In this work, we introduce investigate four namely, anti-totient numbers, half near Zumkeller numbers by using the notion non-coprime residues $n$ including $n$. We formulate propose...

An integer-valued multiplicative function f is said to be polynomially-defined if there a nonconstant separable polynomial $$F(T)\in \mathbb {Z}[T]$$ with $$f(p)=F(p)$$ for all primes p. We study the distribution in coprime residue classes of functions, establishing equidistribution results allowing wide range uniformity modulus q. For example, we show that values $$\phi (n)$$ , sampled over in...

Abstract A set of integers is primitive if it does not contain an element dividing another. Let f ( n ) denote the number maximum-size subsets {1,…,2 }. We prove that limit α = lim n→∞ 1/ exists. Furthermore, we present algorithm approximating with (1 + ε multiplicative error in N steps, showing particular ≈ 1.318. Our can be adapted to estimate all sets {1,…, } as well. address another related...