The Ramanujan Journal

, Volume 42, Issue 1, pp 199–221 | Cite as

A Diophantine approach to the three and four exponentials conjectures

  • Lee A. Butler


Some special cases of Wilkie’s conjecture are shown to be equivalent to real versions of the three and four exponentials conjectures. Wilkie’s conjecture is an open problem originating in model theory that concerns the density of algebraic points in sets defined using the exponential function; the latter conjectures concern the algebraic nature of values of the exponential function.


Transcendence Exponential function Logarithms of algebraic numbers Algebraic independence 

Mathematics Subject Classification

03C64 11J81 



Many thanks to the anonymous referee for numerous improvements to the paper including a substantial strengthening of Theorem 9.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Heilbronn Institute for Mathematical ResearchUniversity of BristolBristolUK

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