Frits Beukers
Name : Frits Beukers
Position : Full Professor
Dept. : Mathematics
Utrecht University
the Netherlands
Office : MI 520
Email : f.beukers at uu.nl
Phone : +31302531419 (work)
Onderwijs/Education
Seminars (local and intercity)
Recent scientific preprints/papers
 F.Beukers,
Consequences of Apery's work on irrationality,
Rencontres
Arithmetiques de Caen, zeta(3) irrationnel: les retombees, 1995

D.Zagier, F.Beukers,
Lower bounds of heights of
points on hypersurfaces
Acta Arithmetica 79(1997), 103111

F. Beukers
Ternary form equations.
J.Number Theory 54 (1995), 113133

H.P.Schlickewei, F.Beukers,
The equation x+y=1
in finitely generated groups
Acta Arithmetica 78(1996), 189199

F. Beukers,
On a sequence of polynomials.
J.pure and applied Algebra 117,118(1997), 97103

F. Beukers
Integral points on cubic surfaces
CRM Proceedings and Lecture Notes 19 (1999), 2533

F. Beukers
The diophantine equation Ax^p+By^q=Cz^r
Duke Math. J. 91 (1998), 6188

R.Cushman, F.Beukers
Zeeman's monotonicity conjecture
Journal of Differential Equations 143(1998), 191200

T.N.Shorey, R.Tijdeman, F.Beukers
Irreducibility
of polynomials and arithmetic progressions with equal products of terms
Proc. Conference in honour of A.Schinzel (eds: Gyory, Iwaniec, Urbanowicz)
W.de Gruyter 1999, 1126

J.Sanders, J.P.Wang, F.Beukers
One symmetry does not
imply integrability
extension of paper which appeared in J.of Differential Equations 146
(1998), 251260.

F.Beukers
On Dwork's accessory parameter problem,
Math.Z. 241 (2002), 425444.
 F.Beukers The maximal differential ideal is generated
by its invariants, Indag. Math. 11 (2000), 1318.
 C.Smyth, F.Beukers Cyclotomic points on curves (16 pages),
Number Theory for the Milennium I, Proceedings of
Milennial Conference, UrbanaChampaign 2000, Volume 1,
p6786, A.K.Peters, 2002
 R.Cushman, F.Beukers, The complex
geometry of the spherical pendulum, Celestial Mechanics,
Contemporary Mathematics 292(2002), 4770.
This version is expanded and corrected from
the original.
 A. van der Waall, F.Beukers
Lame equations with algebraic solutions
(19 pages), J.Differential Equations 197(2004), 125.
 F.Beukers,
A refined version of the SiegelShidlovskii theorem (9 pages,
see also href 0405549),
Annals of Mathematics 163(2006), 369379.
 With H.Montanus: A
compilation of all extremal, semistable
elliptic fibrations of K3surfaces and the associated paper
titled Explicit calculation of elliptic fibrations of K3surfaces and their Belyimaps.
Number theory and polynomials, 3351, London Math. Soc. Lecture Note Ser., 352,
Cambridge Univ. Press, Cambridge, 2008.
 F.Beukers, Irrationality of padic Lvalues,
Acta Mathematica Sinica (English Series) 24, 663686.
 F.Beukers, Unitary monodromy of Lam'e differential operators,
Regul. Chaotic Dyn. 12 (2007), no. 6, 630641.
 Stewart, C. L., Beukers, F.: Neighbouring powers,
J. Number Theory 130 (2010), 660679.
 F.Beukers,
Algebraic Ahypergeometric functions. Invent. Math. 180 (2010), 589610.
 F.Beukers, Irreducibility of Ahypergeometric systems. Indag. Math. (N.S.) 21 (2011), 3039.
 Beukers, Frits; Luca, Florian; Oort, Frans:
Power values of divisor sums. Amer. Math. Monthly 119 (2012), 373380
 F.Beukers, Monodromy of Ahypergeometric functions. This
preprint is a major rewrite of a manuscript posted two years ago.
Recent lectures, semiscientific publications, course notes
 F.Beukers Vakantiecursus 1999: P=NP?
(17 pages, in Dutch)
 F.Beukers A rational approach to Pi
(17 pages), Notes of a lecture held on the occasion of Piday on
July 5, 2000 in Leiden. Nieuw Archief voor Wiskunde 2000, issue 4.
 W.Reinboud, F.Beukers, Snellius versneld
(5 pages, in English), Nieuw Archief voor Wiskunde 3(2002), 6063.
 F.Beukers Experimentele Getaltheorie
(Vakantiecursus 2001, 19 pages in Dutch)
 F.Beukers The Riemann zetafunction
and its relatives" Transparencies from a lecture in the
Basic Notions series of the Math Colloquium at Utrecht on
13 september 2001.
 Oratie gehouden op 22 Oktober 2001.
 With R.M. van Luijk, R.Vidunas,
A linear algebra problem, Nieuw Archief voor Wiskunde 3 (2002),
139140.
 Gauss hypergeometric functions,
Notes from an MRI course given in 1993 and which has slowly developed
into an article of a summerschool held in Istanbul in 2007.
 Hypergeometric functions of one
variable Notes from MRI springschool 1999 held in Groningen
(organisers M.van der Put, J.Top)
 Periods Slides from a lecture held in
Luminy (May 11, 2002) about the algebraic independence of periods
as expounded in the paper Periods by M.Kontsevich and D.Zagier.

Exploring Efunctions, Slides
of a lecture held on June 18, 2004, Waterloo,
on the occasion of W.D.Brownawell's 60th birthday.

The diophantine equation
Ax^p+By^q=Cz^r. Lectures held at Institut Henri Poincare, September 2004.
 The limits of reason Studium generale voordracht,
6 april 2005, naar aanleiding van Hoofdstuk 10 van Peter Atkins' boek
Galileo's finger. In pdf, in Dutch.
 Introduction to the ABCconjecture,
lecture held on September 9, 2005, the kickoff meeting of the project
Reken mee met ABC.
 Waar zijn de reele getallen?,
Demonstration and pictures given during a lecture before the
Nationale Wiskundedagen 2008 on the nature of the real numbers.
In particular is R=?Q. The
complete presentation can also be downloaded (beware, about 30Mb).
 Recurrent sequences coming from Shimura curves,
Lecture given on June 3, 2010 in Banff at the occasion of Cam Stewart's 60th birthday.
 F.Beukers Diophantische vergelijkingen: een
onmogelijke uitdaging (CWI Vakantiecursus 2010, in Dutch)
 F.Beukers tekst van een VWOmasterclass over
kettingbreuken, gehouden op 14 en 15 oktober 2011.
 F.Beukers Notes on Ahypergeometric functions which
appeared in Séminaires et Congrès 23 (2011), 2561,
Société mathématique de France.
 RTV Utrecht
Henk Westbroek en de wiskunde (TV broadcast, in Dutch).
Getaltheorie voor beginners
This is an introduction (written in Dutch) addressed to newcomers in Number
Theory. Complete title:
Frits Beukers,
Getaltheorie voor beginners
Epsilon Uitgaven, Utrecht 1999
ISBN 9050410499
Apart from an elementary introduction a good number of chapters is devoted
to recent developments in elementary number theory. You can find the
pdffile of the Preface and table of contents
here. Of course it is a lot
of fun to perform experiments on a computer. For this reason I collected
some useful software.
Pi
This is an introduction (again in Dutch, about 50 pages)
for high school students into the secrets of PI. It is issue 6
of the ZEBRAseries published jointly by Epsilon Uitgaven
and the Nederlandse Vereniging van Wiskundeleraren. ISBNnumber is
9054010626. You
can find the pdffile of the introduction and table of contents
here. As a result of repeated
requests you can find here the
answers to the exercises in pdfformat.
Research
My research interests are

Diophantine equations

Hypergeometric functions

Number Theory in general (see Number
Theory Web for my colleagues)
Mathematical applets
Other mathematical diversions