turkmath.org

The first proof of the infinitude of prime numbers was offered by Euclid in his Elements in 300 B.C. Then a great number of different proofs of this result, known as Euclid’s Theorem, were given by many mathematicians. A considerable amount of the proofs, including Euclid’s, used the fact that prime numbers are the atoms of positive integers in the multiplicative sense. For instance, using this fact, Leonhard Euler showed the divergence of the sum of reciprocals of all prime numbers, which implies Euclid’s Theorem. In this talk, we will give a novel proof of Euclid’s Theorem using the Polynomial van der Waerden Theorem that is a remarkable result from additive combinatorics. After that, we will also offer a new proof of Euler’s Theorem using some instruments from additive combinatorics. Finally, we will present the Green-Tao Theorem, one of the well-known results in additive combinatorics, for polynomial rings over integral domains with several variables.