On the Number of Monochromatic Triples Associated with Binary Equations over Coloured Algebraic Groups

Schur's Theorem on integer colouring states that colouring integers using finitely many colours yields at least one monochromatic solution to the equation <img src=image/13437836_01.gif>. An extension of Schur's theorem on integer lattices is explored by Vishal Balaji, Andrew Lott and Alex Rice. Schur tried to prove Fermat's Last Theorem by proving non-existence of the solution to the equation <img src=image/13437836_02.gif> for prime <img src=image/13437836_03.gif>. But he in fact proved that "for every integer <img src=image/13437836_04.gif>, there exists <img src=image/13437836_05.gif> such that for any prime <img src=image/13437836_06.gif>, the congruence <img src=image/13437836_02.gif> has a solution" where he failed to prove Fermat's Last Theorem in this route of attack. This demonstrates that Fermat's Last Theorem does not hold in the finite field <img src=image/13437836_07.gif> for any sufficiently large prime <img src=image/13437836_03.gif>. We investigate Schur's Theorem on integer colouring and the corresponding theoretical framework in algebraic groups, and we classify colourings that yield a monochromatic solution to <img src=image/13437836_08.gif> (not all are equal). We use combinatorial tools like bijective counting and Pigeonhole principle to arrive at Theorem 3.9. Our methods include Principle of Inclusion-Exclusion formula to prove the principal result Theorem 3.20. We have used Python language to implement algorithms developed during the research to showcase Schur triples associated to the group <img src=image/13437836_09.gif> and some given colouring maps. We illustrated various groups and its colouring properties. We were able to find bounds using certain parameters involving special subgroups of the group for Schur triples <img src=image/13437836_10.gif> such that <img src=image/13437836_11.gif> and <img src=image/13437836_12.gif> get the same colour in algebraic groups when coloured using finitely many colours. We also find the connection between proper vertex colouring and group colouring via Cayley graphs of semigroups. Our study throws light on new combinatorial perspectives on colouring problems on finite algebraic groups. The results help to enhance new algorithms related to Cayley graph colourings associated with finite semigroups. The research helps in combinatorial studies equipped by colouring problems involving network theory.