Nontrivial Integer Solutions Research Articles

The local solubility for homogeneous polynomials with random coefficients over thin sets

AbstractLet and be natural numbers greater or equal to 2. Let be a homogeneous polynomial in variables of degree with integer coefficients , where denotes the inner product, and denotes the Veronese embedding with . Consider a variety in , defined by . In this paper, we examine a set of integer vectors , defined by where is a nonsingular form in variables of degree with for some constant depending at most on and . Suppose has a nontrivial integer solution. We confirm that the proportion of integer vectors in , whose associated equation is everywhere locally soluble, converges to a constant as . Moreover, for each place of , if there exists a nonzero such that and the variety in admits a smooth ‐point, the constant is positive.

The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Let ψ : R + → R + \psi :\mathbb {R}_+\to \mathbb {R}_+ be a non-increasing function. A real number x x is said to be ψ \psi -Dirichlet improvable if the system | q x − p | > ψ ( t ) and | q | > t \begin{equation*} |qx-p|> \psi (t) \ \ {\text {and}} \ \ |q|>t \end{equation*} has a non-trivial integer solution for all large enough t t . Denote the collection of such points by D ( ψ ) D(\psi ) . In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sublinear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.

Perfect powers that are sums of squares of an arithmetic progression

We determine all nontrivial integer solutions to the equation (x+r)2+(x+2r)2+⋯+(x+dr)2=yn for 2≤d≤10 and 1≤r≤104 with gcd(x,y)=1. We make use of a factorization argument and the primitive divisors theorem due to Bilu, Hanrot and Voutier.

A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

Abstract We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.

A Diophantine equation with the harmonic mean

Let $$f\in \mathbb {Q}[x]$$ be a polynomial without multiple roots and $$\deg {f}\ge 2$$. We give conditions for $$f=x^2+bx+c$$ under which the Diophantine equation $$2f(x)f(y)=f(z)(f(x)+f(y))$$ has infinitely many nontrivial integer solutions and prove that this equation has infinitely many rational parametric solutions for $$f=x^2+bx$$ with nonzero integer b. Moreover, we show that it has a rational parametric solution for infinitely many cubic polynomials.

The Smallest Nontrivial Solution to and Related Sequences

For two positive integers we consider the equation and we let s > 1 be its smallest nontrivial integer solution. We prove that as n ranges over the positive integers, the ratio takes each positive integer value infinitely often.

HAUSDORFF MEASURE OF SETS OF DIRICHLET NON‐IMPROVABLE NUMBERS

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if it admits an improvement to Dirichlet's theorem in the following sense: the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for all large enough $t$. Denote the collection of such points by $D(\psi)$. In this paper, we prove that the Hausdorff measure of the complement $D(\psi)^c$ (the set of $\psi$-Dirichlet non-improvable numbers) obeys a zero-infinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock \& Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

On The Diophantine Equation xa + ya = pkzb

In this study, we consider the Diophantine equation xa + ya = pkzb where p is a prime number, gcd(a, b) = 1 and k,a,b∈Z+. We solve this equation parametrically by considering different cases of x and y and find that there exist infinitely many nontrivial integer solutions, where the formulated parametric solutions solve xa + ya = pkzb completely for the case of x = y, x = −y, and either x or y is zero (not both zero). For the case of |x| ≠ |y| and both x and y nonzero, not every solution (x,y,z) is in the parametric forms proposed in Theorem 5, although any (x,y,z) in these parametric forms solves the Diophantine equation.

RATIONAL POINTS ON CUBIC HYPERSURFACES THAT SPLIT INTO FOUR FORMS

Let be a cubic form. Assume that splits into four forms. Then has a non-trivial integer solution provided that .

A note on the Diophantine equation $$f(x)f(y)=f(z^2)$$ f ( x ) f ( y ) = f ( z 2 )

Let \(f\in {\mathbb {Q}}[X]\) be a polynomial without multiple roots and \(deg(f)\ge 2\). We consider the Diophantine equation \(f(x)f(y)=f(z^2)\). For two classes of irreducible quadratic polynomials, this equation has infinitely many nontrivial integer solutions, if the corresponding Pell’s equations satisfy a condition. For a special cubic polynomial, it has a one-parameter family of rational solutions. For \(f(X)=X(X^2+X+k)\) and \(X(X^2+kX+1)\) there are infinitely many \(k \in {\mathbb {Q}}\) such that the title equation has rational solutions.

On a Diophantine Equation of Stroeker

Abstract : In this paper, we prove that there are infinitely many positive integers N such that the Diophantine equation (x2 + y)(x + y2) = N(x - y)3 has no nontrivial integer solution (x, y).

Sums of two squares and one biquadrate

There are no nontrivial integer solutions of x 2 + y 2 + z 4 = p 2 for primes p 7 (mod 8), even though there are no congruence obstructions.

On a Fermat-Type Diophantine Equation

Let p>3 be an odd prime and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1, (k, p)=1. Let C(i)p be the eigenspace of the ideal class group of Q(ζ) corresponding to ωi, ω being the Teichmuller character. Let B2i denote the 2ith Bernoulli number. In this article we apply the methods (following H. S. Vandiver (1934, Bull. Amer. Math. Soc., 118–123)) which were used by the author (1994, Ph.D. Thesis, California Institute of Technology) to prove a special case Fermat's Last theorem, to study the equation xp+yp=pczp. In particular, we prove the following: Assume p is irregular, and p∣Bp−3. Let q be an odd prime such that q≡1 (modp), and there is a prime ideal Q over q in Q(ζ) whose ideal class generates C(3)p, which is known to be cyclic. If xp+yp=pczp has nontrivial integer solutions, then we show that q∤(pczp/(x+y)). We also give proof of the unsolvability of the above equation for regular primes (p>3), using the results of H. S. Vandiver (1936, Monatsh. Math. Phys.43, 317–320).

An Elementary Solution of Lucas′ Problem

By the method of infinite descent, Pell′s equation, and the quadratic reciprocity law, it is proved that the equation x( x + 1)(2 x + 1) = 6 y 2 has the only nontrivial integer solution x = 24, y = 70.