Related papers: Prime Solutions of Diagonal Diophantine Systems

We explicitly construct a diffeomorphic pair (p(x),p^{-1}(x)) in terms of an appropriate quadric spline interpolating the prime series. These continuously differentiable functions are the smooth analogs of the prime series and the prime…

By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and…

Let a1,..., a9 be non-zero integers and n any integer. Suppose that a1 + ... + a9 = n (mod 2) and (ai, aj) = 1 for 1 <= i < j <= 9. We will prove that (i) if not all of the aj's are of the same sign, then the cubic diagonal equation a1p1^3…

We consider equations of the form $a_{1}x_{1}^{k}+...+a_{s}x_{s}^{k}$ and when they have solutions in the primes. We define an analogue of the Hasse principle for solubility in the primes (which we call the prime Hasse principle), and prove…

In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine…

The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our…

This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood.…

Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients,…

In this paper we prove an asymptotic formula for the number of solutions in prime numbers to systems of simultaneous linear inequalities with algebraic coefficients. For $m$ simultaneous inequalities we require at least $m+2$ variables,…

Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. Suppose that $\boldsymbol{F}$ is a nonsingular…

We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ \theta ,$ where $ \theta >2$ is real and non-integral. For fixed non-zero real numbers $ \lambda_i $ not all of the same sign we write…

In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on…

We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the…

The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of DP1 are classified under criterion of their behavior while argument tends to infinity. The Isomonodromic Deformations Method yields asymptotic…

Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n) = 0 \ (1…

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring's problem. In particular, we show that the anticipated asymptotic formula in Waring's problem for sums of…

Let $1<k<7/6$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality…

In this paper we show that, for any fixed $1<c<967/805$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon \end{equation*}…

Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $\alpha$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(\alpha)/\mathbb Q$ is Galois. We prove that, for every…