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

This article is cited in 18 scientific papers (total in 18 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

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
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    6. Hessami Pilehrood Kh., Hessami Pilehrood T., “Infinite sumas as linear combinations of polygamma functions”, Acta Arith., 130:3 (2007), 231–254  crossref  mathscinet  zmath  isi  scopus
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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
    10. J. Cassaigne, F. V. Petrov, A. E. Frid, “On possible growths of Toeplitz languages”, Siberian Math. J., 52:1 (2011), 63–73  mathnet  crossref  mathscinet  isi
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    14. Arakawa T., Ibukiyama T., Kaneko M., “Bernoulli Numbers and Zeta Functions”, Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, 2014, 1–274  crossref  mathscinet  isi
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    18. Bugeaud Y., “Linear Forms in Logarithms and Applications”, Linear Forms in Logarithms and Applications, Irma Lectures in Mathematics and Theoretical Physics, 28, European Mathematical Soc, 2018, 1–224  crossref  mathscinet  isi
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