Number theory is one of the oldest branches of mathematics, and even more alive than ever before. The subject holds an attraction to both mathematics amateurs and professional mathematicians.
Two subjects of popular attraction are the so-called Money changing problem and the 3n+1-problem. Click on the links to play with the corresponding applets.
But of course also Fermat's Last Theorem still has its attraction. This was witnessed by the more than 500 visitors (teachers, high school students, other mathematically interested) of the Fermat day held in Utrecht on November 6, 1993. There the introductory and more advanced aspects of Wiles' work on the solution of Fermat's last problem were presented.
As one knows Fermat's Last Theorem asserts that, given any integer n>2, the sum of two positive n-th powers cannot be an n-th power. Fermat raised this problem around 1650, and it was only proved by Andrew Wiles in 1994. Actually, Wiles proved the much more profound Shimura-Taniyama-Weil conjecture, of which Fermat's Theorem was merely a consequence (this connection was not noticed until 1986). For a popular account of this story see the beautiful book by Simon Singh.
A generalisation of Fermat's problem is of course the diophantine equation xp+yq=zr in coprime integers x,y,z. A theorem by Darmon and Granville states that if p,q,r are fixed integers such that 1/p+1/q+1/r< 1 (the so-called hyperbolic case), then the number of solutions x,y,z is finite. During the preparation of the Fermat day it was noticed by F.Beukers and D.Zagier that beside the known solutions 23+15= 32, etc there are a number of surprisingly large solutions. The complete list of known hyperbolic-type solutions is:
| 1k + 23 = 32 (k>6) | 132 + 73 = 29 | 27 + 173 = 712 |
| 25 + 72 = 34 | 35 + 114 = 1222 | 177 + 762713= 210639282 |
| 14143 + 22134592= 657 | 338 + 15490342 = 156133 | 438 + 962223 = 300429072 |
| 92623 + 153122832= 1137 |
When 1/p+1/q+1/r>1 we are in the so-called spherical case. The relevant sets {p,q,r} are {2,2,k}, {2,3,3}, {2,3,4}, {2,3,5}. In all these cases it is known that the equation has infinitely many solutions. For most of these cases the solution is tedious but straightforward, but the equation x5+ y3= z2 remained untouched until it was solved by work of F.Beukers (1998) and Johnny Edwards (2000), an Utrecht PhD-student. For a recent course on the subject see The Catalan-Fermat equation.
Another topic of popular interest is irrationality and transcendence of numbers. During the lecture Transcendental Numbers an account for a general mathematical audience was given. The Utrecht involvement (F.Beukers) in the subject has been through the alternative irrationality proof of zeta(3) in 1979 and the answer (2004) to a long-standing question of S.Lang on the transcendence of so-called E-functions (see the slides of the lecture).
Number theory has many connections with other branches of mathematics, it fits seamlessly with (arithmetic) algebraic geometry but also the subject of linear differential equations contains many questions of arithmetic and algebraic interest.
As an example of differential equations with arithmetic
interest we mention the hypergeometric equation. The classification of one-variable
hypergeometric equation with algebraic solutions was started in 1873 by H.A.Schwarz
for classical second order equations and finished in 1987 for general order
equations by F.Beukers and G.Heckman. For an introduction to hypergeometric
functions one can consult the lecture notes of courses given in the past:
Gauss hypergeometric
functions and One variable
hypergeometric functions.
Above one sees 36 marbles arranged
in both a triangular and a square pattern. The question is which quantities
of marbles can be arranged both as a square and a triangle. Denote such a
quantity by n. Of course n=1 is a possible (trivial) solution.
Perhaps surprisingly
it turns out that there infinitely many solutions n. Here are the first few
values
1, 36, 1225, 41616, ...
It is not hard to see that determination of them comes
down to solving a simple Pellian equation.
The Schwarz map of a linear second order differential equation is simply given by the quotient of two independent solutions. On the left we depicted the action of a Schwarz map belonging to a hypergeomtric equation with real parameters. It typically maps the complex upper half plane conformally onto a triangelewith circle segments as edges.