898 research outputs found
Diophantine m-tuples for quadratic polynomials
In this paper, we prove that there does not exist a set with more than 98 nonzero polynomials in Z[X], such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z[X] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial pZ[X] such that p2|n). Specially, we prove that if such a set contains only polynomials of odd degree, then it has at most 18 elements
Report on some recent advances in Diophantine approximation
A basic question of Diophantine approximation, which is the first issue we
discuss, is to investigate the rational approximations to a single real number.
Next, we consider the algebraic or polynomial approximations to a single
complex number, as well as the simultaneous approximation of powers of a real
number by rational numbers with the same denominator. Finally we study
generalisations of these questions to higher dimensions. Several recent
advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent,
T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these
works.Comment: to be published by Springer Verlag, Special volume in honor of Serge
Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and
John Tat
Open Diophantine Problems
We collect a number of open questions concerning Diophantine equations,
Diophantine Approximation and transcendental numbers. Revised version:
corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1
(2004) dedicated to Pierre Cartie
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