Fragments of the book “The Great Mathematical Problems”, by Ian Stewart.
Goldbach Conjecture
- Question: Can every even number greater than 2 be written as the sum of two primes?
- Formulation: On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler
- Status: it holds but has not been proved yet. It is one of the oldest and best-known unsolved problems in number theory and all of mathematics
Squaring the Circle
- Question: It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge
- Formulation: a problem in geometry first proposed in Greek mathematics
- Status: In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ({\displaystyle \pi }\pi ) is a transcendental number. That is, {\displaystyle \pi }\pi is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if {\displaystyle \pi }\pi were transcendental, but that fact was not proven until 1882.
The Four Colour Theorem
- Question: no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet
- Formulation: the conjecture was first proposed on October 23, 1852, when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed.
- Status: It was the first major theorem to be proved using a computer.
Kepler Conjecture
- Question: the densest packing of three-dimensional Euclidean space by equal spheres is attained by the “cannonball” packing
- Formulation: one of geometry’s oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list
- Status: this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers.
Mordell Conjecture
- Question: a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points
- Formulation: in arithmetic geometry, conjecture made by Louis Mordell
- Status: in 1983 it was proved by Gerd Faltings, and is now known as Faltings’s theorem. The conjecture was later generalized by replacing Q by any number field.