Partition regularity of $x+y=z^2$ over $\mathbb{Z}/p\mathbb{Z}$

15 February 2016
Sofia Lindqvist

Consider the following question. Given a $k$-colouring of the positive integers, must there exist a solution to $x+y=z^2$ with $x,y,z$ all the same colour (and not all equal to 2)? Using $10$ colours a counterexample can be given to show that the answer is "no". If one instead asks the same question over $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$, the answer turns out to be "yes", provided $p$ is large enough in terms of the number of colours used.  I will talk about how to prove this using techniques developed by Ben Green and Tom Sanders. The main ingredients are a regularity lemma, a counting lemma and a Ramsey lemma.

  • Junior Number Theory Seminar