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Mosc. Math. J., 2004, Volume 4, Number 1, Pages 245–305 (Mi mmj150)  

This article is cited in 17 scientific papers (total in 17 papers)

Open Diophantine problems

M. Waldschmidt

Institut de Mathématiques de Jussieu

Abstract: 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.

Key words and phrases: Diophantine problems, transcendence, linear and algebraic independence, Schanuel conjecture, four exponential conjecture, multizeta values, $abc$ conjecture, Waring problem, Diophantine approximation, continued fractions, Thue–Siegel–Roth–Schmidt, irrationality measures, Markoff spectrum, height, Lehmer problem, Mazur density conjecture, abelian varieties, special functions, function fields.

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MSC: Primary 11Jxx; Secondary 11Dxx, 11Gxx, 14Gxx
Received: October 16, 2002
Language: English

Citation: M. Waldschmidt, “Open Diophantine problems”, Mosc. Math. J., 4:1 (2004), 245–305

Citation in format AMSBIB
\by M.~Waldschmidt
\paper Open Diophantine problems
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 1
\pages 245--305

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    2. Sicard A., Ospina J., Velez M., “Numerical simulations of a possible hypercomputational quantum algorithm”, Adaptive and Natural Computing Algorithms, Springer Computer Science, 2005, 272–275  isi
    3. Fischler S., Rivoal T., “Un exposant de densité en approximation rationnelle [A density exponent in rational approximation]”, Int. Math. Res. Not., 2006, 95418, 48 pp.  mathscinet  zmath  isi  elib
    4. Waldschmidt M., “Transcendence of periods: The state of the art”, Pure Appl. Math. Q., 2:2 (2006), 435–463  crossref  mathscinet  zmath  isi
    5. Corvaja P., Zannier U., “On the greatest prime factor of Markov pairs”, Rend. Sem. Mat. Univ. Padova, 116 (2006), 253–260  mathscinet  zmath  isi
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    9. Bugeaud Ya., Reutenauer Ch., Siksek S., “A Sturmian sequence related to the uniqueness conjecture for Markoff numbers”, Theoret. Comput. Sci., 410:30–32 (2009), 2864–2869  crossref  mathscinet  zmath  isi  scopus
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