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Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Transcendence of Periods: The State of the Art
, 2005
"... function, multiple zeta values (MZV). AMS subject classifications 11J81 11J86 11J89 Abstract: The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are “interesting”, which appear “naturally”, which deserve our attention, form a countabl ..."
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Cited by 8 (0 self)
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function, multiple zeta values (MZV). AMS subject classifications 11J81 11J86 11J89 Abstract: The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are “interesting”, which appear “naturally”, which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M. Kontsevich and D. Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers. Acknowledgment This is an english updated version of the paper in french: Transcendance de périodes: état des connaissances,
Composition factors from the group ring and Artin's theorem on orders of simple groups
 Proc. London Math. Soc
, 1990
"... The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically si ..."
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Cited by 8 (2 self)
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The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups. 1.
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.
1. Historical introduction to irrationality
, 2007
"... Irrationality of π by Lambert. Continued fraction expansion of e by Euler. 1.2 Fourier. Fourier’s proof of the irrationality of e. Extensions by Liouville. 1.3 Introduction to Hermite’s work. Proof of the irrationality of e r for r a nonzero rational number. Proof of the irrationality of π. Proof o ..."
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Irrationality of π by Lambert. Continued fraction expansion of e by Euler. 1.2 Fourier. Fourier’s proof of the irrationality of e. Extensions by Liouville. 1.3 Introduction to Hermite’s work. Proof of the irrationality of e r for r a nonzero rational number. Proof of the irrationality of π. Proof of the irrationality of e r ̸ ∈ Q(i) for r ∈ Q(i) ×. 2. Historical introduction to transcendence methods.
The main reference is Nesterenko’s recent book [11].
, 2010
"... Proposition 1. Let ϑ be a real number. The following conditions are equivalent (i) ϑ is irrational. (ii) For any ɛ> 0, there exists p/q ∈ Q such that ..."
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Proposition 1. Let ϑ be a real number. The following conditions are equivalent (i) ϑ is irrational. (ii) For any ɛ> 0, there exists p/q ∈ Q such that
3.3 Linear forms
, 2010
"... 3.3.1 Siegel’s method: m +1 linear forms For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simult ..."
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3.3.1 Siegel’s method: m +1 linear forms For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coefficients by H(L) = max{a0,..., am}. Lemma 17. Let ϑ1,..., ϑm be complex numbers. Assume that, for any ɛ> 0, there exists m +1 linearly independent linear forms L0,..., Lm in m +1 variables, with coefficients in Z, such that max 0≤k≤m Lk(1, ϑ1,..., ϑm)  < ɛ where H = max
2.3 Linear forms 2.3.1 Siegel’s method: m +1 linear forms
"... For proving linear independence of real numbers, Hermite [6] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [14] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independe ..."
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For proving linear independence of real numbers, Hermite [6] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [14] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coefficients by H(L) = max{a0,..., am}. Lemma 13. Let ϑ1,..., ϑm be complex numbers. Assume that, for any ɛ> 0, there exists m +1 linearly independent linear forms L0,..., Lm in m +1 variables, with coefficients in Z, such that max 0≤k≤m Lk(1,ϑ1,..., ϑm)  < ɛ where H = max
Pythagorean triples, rational angles, and spacefilling simplices
, 2003
"... The ancient Greeks posed and solved the problem of finding all right triangles with rational sidelengths. There are 4 natural nonEuclidean generalizations of this problem. We solve them all. The result is that the only rationalsided nonEuclidean triangle with one right angle is the isoceles spheric ..."
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The ancient Greeks posed and solved the problem of finding all right triangles with rational sidelengths. There are 4 natural nonEuclidean generalizations of this problem. We solve them all. The result is that the only rationalsided nonEuclidean triangle with one right angle is the isoceles spherical triangle with legs of length 45 ◦ and hypotenuse 60 ◦. We next ask which simplices have rational dihedral angles (measured in degrees). The solution is easy once connection is made to 1934 work of Coxeter. There are only a finite number of examples in 3dimensional Euclidean space and only a countable number in nspace for n ≥ 4, which are nowhere dense in the space of simplices. But there are a dense and infinite set of examples if n = 2 or in nonEuclidean nspaces for each n ≥ 2. In contrast, there are a continuum infinity of nsimplex shapes which tile nspace and are equidecomposable with nparallelipipeds, as we show