On a Popular Diophantine Norm Equation

Formalized Mathematics, 2017

SummaryIn the article we present in the Mizar system [1], [2] the formalized proofs for Hurwitz’ theorem [4, 1891] and Minkowski’s theorem [5]. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in [3], [6]. A formal proof of Dirichlet’s theorem, namely an inequation |θ−y/x| ≤ 1/x2has infinitely many integer solutions (x, y) where θ is an irrational number, was given in [8]. A finer approximation is given by Hurwitz’ theorem: |θ− y/x|≤ 1/√5x2. Minkowski’s theorem concerns an inequation of a product of non-homogeneous binary linear forms such that |a1x + b1y + c1| · |a2x + b2y + c2| ≤ ∆/4 where ∆ = |a1b2− a2b1| ≠ 0, has at least one integer solution.

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.

Rocky Mountain Journal of Mathematics, 2006

Journal of Number Theory, 2011

The title equation, where p > 3 is a prime number ≡ 7 (mod 8), q is an odd prime number and x, y, n are positive integers with x, y relatively prime, is studied. When p ≡ 3 (mod 8), we prove (Theorem 2.3) that there are no solutions. For p ≡ 3 (mod 8) the treatment of the equation turns out to be a difficult task. We focus our attention to p = 5, by reason of an article by F.

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.

Proceedings of the National Academy of Sciences, 1984

Siegel&#39;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 &gt; 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. &...

Journal of Number Theory, 2005

Let D > 2 be a positive integer, and let p be an odd prime not dividing D. In this paper, using the deep result of Bilu, Hanrot and Voutier (i.e., the existence of primitive prime factors of Lucas and Lehmer sequences), by computing Jacobi's symbols and using elementary arguments, we prove that: if (D, p) = (4, 5), (2, 5), then the diophantine equation x 2 + D m = p n has at most two positive integer solutions (x, m, n). Moreover, both x 2 + 4 m = 5 n and x 2 + 2 m = 5 n have exactly three positive integer solutions (x, m, n).

Journal of Number Theory, 2014

This note presents corrections to the paper by Y. Wang and T. Wang [2]. The unique theorem given in that paper states that for any odd integer n > 1, nx 2 + 2 2m = y n has no positive integer solution (x, y, m) with gcd(x, y) = 1.

Acta Arithmetica, 2014

In this note we present some results concerning the unirationality of the algebraic variety S f given by the equation

Transactions of the American Mathematical Society, 1966

Introduction. The study of the values at rational points of transcendental functions defined by linear differential equations with coefficients in Q[z] (2) can be traced back to Hurwitz [1] who showed that if ,. , 1 z 1 z2 Az) = l+-b-lT+WTa)2l +where « is a positive integer, b is an integer, and b\a is not a negative integer, then for all nonzero z in Q((-1)1/2) the number y'(z)jy(z) is not in g((-1)1/2). Ratner [2] proved further results. Then Hurwitz [3] generalized his previous results to show that if nZ) ,+g(0) 1! +g(0)-g(l)2! + where f(z) and g(z) are in Q[z\, neither f(z) nor g(z) has a nonnegative integral zero, and degree (/(z)) < degree (g(z)) = r, then for all nonzero z in the imaginary quadratic field Q((-n)1'2) two of the numbers y(z),y(l)(z),-,yir\z) have a ratio which is not in Q((-n)112). Perron [4], Popken [5], C. L. Siegel [6], and K. Mahler [7] have obtained important results in this area. In this paper we shall use a generalization of the method which was developed by Mahler [7] to study the approximation of the logarithms of algebraic numbers by rational and algebraic numbers. Definition. Let K denote the field Q((-n)i/2) for some nonnegative integer «. Definition. For any monic 0(z) in AT[z] of degree k > 0 and such that 6(z) has no positive integral zeros we define the entire function oo d f(z)= £-. d^O d n oc«)