### Partition regularity of Pythagorean pairs

Heilbronn Number Theory Seminar

13th December 2023, 4:00 pm – 5:00 pm

Fry Building, 2.04

Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x^2+y^2=z^2)? This is one of the simplest (to state!) questions in arithmetic Ramsey theory which is still widely open. I will present a recent partial result, showing that "Pythagorean pairs" are partition regular, that is in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x^2+y^2=z^2 for some integer z (which may be coloured differently). The proof is a blend of ideas from ergodic theory and multiplicative number theory. Based on a joint work with N. Frantzikinakis and J. Moreira.

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