We show that Artin’s conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r > 2.

Abstract In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue Khintchine’s theorem for these sets. We then apply to the topic intrinsic Diophantine Approximation on self-similar particular, define new height element $${\mathbb {Q}}^d$$ <mml:m...

We generalize Khintchine’s method of constructing totally irrational singular vectors and linear forms. The main result the paper shows existence forms with large uniform Diophantine exponents on certain subsets ℝn, in particular any analytic submanifold ℝn dimension ≥2 which is not contained a proper rational affine subspace.

The Greek mathematician Diophantus of Alexandria noted that the numbers x, x + 2, 4x + 4 and 9x + 6, where x = 1 16 , have the following property: the product of any two of them increased by 1 is a square of a rational number (see [4]). Fermat first found a set of four positive integers with the above property, and it was {1, 3, 8, 120}. Later, Davenport and Baker [3] showed that if d is a posi...

A set {a1, . . . , am} of m distinct positive integers is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all i, j with 1 ≤ i < j ≤ m. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a, b, c} with a < b < c, the number of Diophantine qui...

The paper describes an automatic tuning procedure for a wide class of continuous-time systems without and with time delays. Every auto-tuning method has two basic steps. The first one is experimental and yields an estimated model. The second step consists in a design procedure of controllers. The developed auto-tuning procedure identifies a first order estimation model obtained by a biased rela...

We describe a new algorithm for solving a conjunction of linear diophantine equations, inequations and disequations in natural numbers. We derive our algorithm from one proposed by Elliott in 1903 for solving a single homogeneous equation. This algorithm was then extended to solve homogeneous systems of equations by MacMahon. We show how it further extends to an algorithm which solves general l...