Simultaneous Diophantine Approximation with Primes

Journal of the London Mathematical …, 1982

THEOREM 1. Suppose that Xx, X2, X3 are non-zero real numbers not all of the same sign, that n is real, and that Xx/X2 is irrational. Let 6 > 0 be given. Then there are infinitely many ordered triples of primes px,p2, p3for which ... THEOREM 2. Given the hypotheses of Theorem 1 ...

Proceedings of the National Academy of Sciences, 1984

Siegel's results [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1] on the transcendence and algebraic independence of values of E -functions are refined to obtain the best possible bound for the measures of irrationality and linear independence of values of arbitrary E -functions at rational points. Our results show that values of E -functions at rational points have measures of diophantine approximations typical to “almost all” numbers. In particular, any such number has the “2 + ε” exponent of irrationality: ǀΘ - p / q ǀ > ǀ q ǀ -2-ε for relatively prime rational integers p,q , with q ≥ q 0 (Θ, ε). These results answer some problems posed by Lang. The methods used here are based on the introduction of graded Padé approximations to systems of functions satisfying linear differential equations with rational function coefficients. The constructions and proofs of this paper were used in the functional (nonarithmetic case) in a previous paper [Chudnovsky, D. V. &...

Formalized Mathematics, 2015

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

Journal of Number Theory, 1986

Glasgow Mathematical Journal, 1996

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequalityHere

Proceedings of the National Academy of Sciences, 1984

This paper is devoted to the study of the arithmetic properties of values of G -functions introduced by Siegel [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1]. One of the main results is a theorem on the linear independence of values of G -functions at rational points close to the origin. In this theorem, no conditions are imposed on the p -adic convergence of a G -function at a generic point. The theorem finally realizes Siegel's program on G -function values outlined in his paper.

By applying Schmidt's lattice method, we prove results on simultaneous Diophantine approximation modulo 1 for systems of polynomials in a single prime variable provided that certain local conditions are met.

Acta Arithmetica, 2006

International Journal of Mathematics and Mathematical Sciences, 2005

We prove the existence of a dense subset ∆ of [0,4] such that for all α ∈ ∆ there exists a subgroup X α of infinite rank of Z[z] such that X α is a discrete subgroup of C[0,β] for all β ≥ α but it is not a discrete subgroup of C[0,β] for any β ∈ (0,α).

2012

We study the distribution of the values of the form λ 1 p 1 + λ 2 p 2 + λ 3 p k 3 , where λ 1 , λ 2 and λ 3 are non-zero real number not all of the same sign, with λ 1 /λ 2 irrational, and p 1 , p 2 and p 3 are prime numbers. We prove that, when 1 < k < 4/3, these value approximate rather closely any prescribed real number.

Rocky Mountain Journal of Mathematics, 2006

Journal of Number Theory, 2012

Let q be a nonzero rational number. We investigate for which q there are innitely many sets consisting of ve nonzero rational numbers such that the product of any two of them plus q is a square of a rational number. We show that there are innitely many square-free such q and on assuming the Parity Conjecture for the twists of an explicitly given elliptic curve we derive that the density of such q is at least one half. For the proof we consider a related question for polynomials with integral coecients. We prove that, up to certain admissible transformations, there is precisely one set of non-constant linear polynomials such that the product of any two of them except one combination, plus a given linear polynomial is a perfect square.

Journal of the London Mathematical Society, 1990

We nd in nitely many pairs of coprime integers, a and q, such that the least prime a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators.

ACTA ARITHMETICA-WARSZAWA-, 2006

Mathematics and Statistics, 2022

In this work we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the form P = (10x + 9)(10y + 9) or P = (10x + 1)(10y + 1) or P = (10x + 7)(10y + 3) where P is a given natural number that ends in one. This allows us to decide the primality of a natural number P that ends in one. Also we get some results on twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension n, n ≥ 2 , in which the relation is obtained: a 1 + a 2 + • • • + a n = λ[a 2 1 + • • • + a 2 n ], where a i is a real number for i = 1, ..., n. When n = 3 or n = 2 we manage to address Fermat's Last Theorem and the equation x 4 + y 4 = z 4 , proving that both equations do not have positive integer solutions. For n = 2, the Cauchy-Schwartz Theorem and Young's inequality were proved in an original way.