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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.

Full text: http://www.ams.org/.../abst4-1-2004.html
References: PDF file   HTML file

Bibliographic databases:

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
\Bibitem{Wal04}
\by M.~Waldschmidt
\paper Open Diophantine problems
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 1
\pages 245--305
\mathnet{http://mi.mathnet.ru/mmj150}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2074991}
\zmath{https://zbmath.org/?q=an:1066.11030}


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    This publication is cited in the following articles:
    1. Vidūnas R., “Expressions for values of the gamma function”, Kyushu J. Math., 59:2 (2005), 267–283  crossref  mathscinet  zmath  isi  scopus
    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
    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
    7. Waldschmidt M., “Transcendance of periods: state of cognition”, Proceedings of the Tunisian Mathematical Society, 2007, 89–116  mathscinet  isi
    8. Dujella A., Petricevic V., “Strong diophantine triples”, Experiment. Math., 17:1 (2008), 83–89  crossref  mathscinet  zmath  isi  scopus
    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
    11. Murty M.R., Weatherby Ch.J., “On the Transcendence of Certain Infinite Series”, Int J Number Theory, 7:2 (2011), 323–339  crossref  mathscinet  zmath  isi  scopus
    12. Philippon P., “Functional Approximations of Projective Space Curbs”, Int J Number Theory, 7:5 (2011), 1195–1215  crossref  mathscinet  zmath  isi  scopus
    13. Ghosh A. Gorodnik A. Nevo A., “Diophantine Approximation and Automorphic Spectrum”, Int. Math. Res. Notices, 2013, no. 21, 5002–5058  crossref  mathscinet  zmath  isi  scopus
    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
    15. Murty M.R., Weatherby Ch., “Special Values of the Gamma Function At Cm Points”, Ramanujan J., 36:3 (2015), 355–373  crossref  mathscinet  zmath  isi  scopus
    16. Luca F., Pizarro-Madariaga A., Pomerance C., “on the Counting Function of Irregular Primes”, Indag. Math.-New Ser., 26:1 (2015), 147–161  crossref  mathscinet  zmath  isi  scopus
    17. Lelis J., Marques D., Ramirez J., “A Note on Transcendental Entire Functions Mapping Uncountable Many Liouville Numbers Into the Set of Liouville Numbers”, Proc. Jpn. Acad. Ser. A-Math. Sci., 93:9 (2017), 111–114  crossref  zmath  isi  scopus
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