One of the famous theorems of Fermat tells us that every prime number giving a remainder 1 while dividing by 4 can be expressed as a sum of two squares of natural numbers. During the talk the history of the theorem will be presented along with two proofs of it. The first of it uses the pigeonhole principle and the fact that -1 is a quadratic residue modulo p then p is a prime number of the form p = 4k + 1. The second of proofs presented, given by H.J.S. Smith, makes use of the continued fractions theory.
To understand the talk only elementary knowledge is required, as the talker will provide all definitions necessary. However, basic knowledge of the congruences would be welcome.