Everyone knows about numbers, don’t they? In fact, “normal people” have more faith in numbers than the mathematicians themselves, who for a long time have racked their brains over the philosophical basis of numbers. Traditionally, mathematicians have also been proud of how their discipline in its pure form is practically useless and have left the invention of practical applications and calculation methods to engineers and scientists of many other fields.

There is, however, a particular branch of mathematics, number theory, which deeply addresses and studies the properties of numbers. The prime numbers are well- known: the numbers 2, 3, 5, 7, 11, and so on. There is no end to the series of primes.

Number theorists have for a long time tried to find a method to calculate them directly, instead of systematically testing all numbers to discover them. Today one does not even check all numbers, but only special candidates for which there is a relatively fast method to test for primality. The largest prime known today is a number with more than 24 million digits, but a new record is set almost every year. By an analogy, the primes are often called “the atoms of mathematics” because they are the constituents of all numbers. Sometimes it is difficult to find a proof based on simple axioms about numbers and mathematicians have tried for centuries without success. “Fermat’s last theorem” is one of the most famous examples: an assertion made in a simple form in the 17th century by Pierre de Fermat, it was proved only in 1994 by Andrew Wiles (so it is now sometimes called the “Fermat-Wiles theorem”).

Simple questions about numbers sometimes arise from tasks in popular books aimed at presenting mathematics to non-specialists in an entertaining way. Readers often have to think up their own method for solving a problem. When the mathematical amateur is not content with just solving calculation problems (even when using a method they invent for themselves), but raises further questions, it gets more interesting. I will present several examples of that.