Let $k\ge 5$ be an odd integer and $\eta$ be any given real number. We prove that if ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, ${\lambda }_4$, $\mu$ are nonzero real numbers, not all of the same sign, and ${\lambda }_1/{\lambda }_2$ is irrational, then for any real number $\sigma$ with $0<\sigma <1/\left(8\vartheta \right(k\left)\right)$, the inequality $$|{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^2+{\lambda }_3{p}_3^2+{\lambda }_4{p}_4^2+\mu p_5^k+\eta |<{\left(\underset{1\le j\le 5}{max}{p}_j\right)}^{-\sigma }$$ has infinitely many solutions in prime variables $p_1,p_2,\cdots ,p_5$, where $\vartheta \left(k\right)=3\times 2^{(k-5)/2}$ for $k=5,7,9$ and $\vartheta \left(k\right)=[(k^2+2k+5)/8]$ for odd integer $k$ with $k\ge 11$. This improves a recent result in W. Ge, T. Wang (2018).

Let 1 < c < 10/9. For large real numbers R > 0, and a small constant η > 0, the inequality $|p{₁}^c+p{₂}^c+p{₃}^c-R|<R^{-\eta }$ holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].

Algebraic bounds of Fréchet classes of copulas can be derived from the fundamental attributes of the associated copulas. A minimal system of algebraic bounds and related basic bounds can be defined using properties of pointed convex polyhedral cones and their relationship with non-negative solutions of systems of linear homogeneous Diophantine equations, largely studied in Combinatorics. The basic bounds are an algebraic improving of the Fréchet-Hoeffding bounds. We provide conditions of compatibility...

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. "$d$-th root problem"), which in short asks whether the integrality of the values of the quotient (resp. $d$-th root) of two (resp. one) linear recurrences implies that this quotient (resp. $d$-th root) is itself a recurrence. We shall also relate such...

The ring of power sums is formed by complex functions on $\mathbb{N}$ of the form$$\alpha \left(n\right)=b_1{c}_1^n+b_2{c}_2^n+...+b_h{c}_h^n,$$for some $b_i\in \overline{\mathbb{Q}}$ and $c_i\in \mathbb{Z}$. Let $F(x,y)\in \overline{\mathbb{Q}}[x,y]$ be absolutely irreducible, monic and of degree at least $2$ in $y$. We consider Diophantine inequalities of the form$$\left|F(\alpha \left(n\right),y)\right|\&lt;|\frac{\partial F}{\partial y}(\alpha \left(n\right),y)|·{\left|\alpha \left(n\right)\right|}^{-\epsilon }$$and show that all the solutions $(n,y)\in \mathbb{N}\times \mathbb{Z}$ have $y$ parametrized by some power sums in a finite set. As a consequence, we prove that the equation$$F\left(\alpha \right(n),y)=f\left(n\right),$$with $f\in \mathbb{Z}\left[x\right]$ not constant, $F$ monic in $y$ and $\alpha$ not constant, has only finitely many solutions.