Classical problems

The old Greeks were unable to find a method for trisecting arbitrary angles with only a compass and a ruler. After a couple of thousand years, in particular towards the end of the 1800s, one could finally prove that angle trisection with compass and ruler is not possible in general. The proof involves both abstract algebra and mathematical analysis. Similar techniques from algebra and so-called group theory made our famous and beloved Niels Henrik Abel proved in 1824 that the general quintic equation did not have a solution in radicals, in contrast to, for example, the solution of quadratic equations.

Possibly the most well-known mathematical problem in history is Fermat's last theorem. In 1637, the French mathematician Pierre de Fermat formulated the following theorem:

If an integer n is larger than 2, then the equation aⁿ + bⁿ = cⁿ has no solutions among the positive integers a, b, and c.

Fermat wrote, in his own edition of the Arithmetica of Diophantus (see the image), "I have discovered a truly marvellous proof [of this theorem]. This margin is too narrow to contain it." After Fermat's lifetime, more and more mathematicians tried to find a proof for this theorem, but it proved to be difficult. The problem, although easy to formulate, developed into one of the most intricate and enigmatic problems that has ever been formed. An enormous number of false proofs the this theorem have been produced. A correct proof, which involved ideas from several fields in pure mathematics, such as number theory, algebra and algebraic geometry, was found after 357 years, in 1995, by the English mathematician Sir Andrew Wiles.

In the year 1900 Hilbert presented to the International Congress of Mathematicians and list of 24 unsolved mathematical problems. These problems shaped mathematics in the twentieth century. Most of them are now solved, however, the Riemann hypothesis, which implies a connection between the distribution of prime numbers and the zeros of a particular function remains the most important unsolved mathematical problem.

In the year 2000, the Clay Mathematics Institute released a list of seven great unsolved mathematical problems. This time with a prize of one million U.S. dollars for the solution of each problem. Six of these problems remain unsolved, however the Poincaré conjecture was solved by the Russian mathematician Grigori Perelman in 2003. In short, the conjecture is as follows: The 2-sphere has the nice property that an elastic band around it can be reduced to a point without either the sphere or the elastic band breaking. This is not the case with an elastic band around a torus, the mathematical concept of a "doughnut". One can show that the 2-sphere (and deformations thereof) are the only two-dimensional surface with the property that an elastic band around it can be reduced to a point without either the surface or the elastic band breaking. The French mathematician Henri Poincaré asked in 1904 whether the 3-sphere consisting of all points equidistant from the origin in a four-dimensional Euclidean space (and deformations thereof) is the only three-dimensional surface where every elastic band can be reduced to a point without breaking. Perelman showed that the answer is "Yes". The proof contains ideas from several field in pure mathematics, such as analysis, topology and differential geometry.