Partition regularity of generalised Fermat equations

2016/06/23 by Lindqvist, Sofia
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1606.07334

Abstract

Let α,β,γ∈ℕ. We prove that given an r-colouring of \mathbbF_p with p prime, there are more than c_r,α,β,γ p² solutions to the equation x^α+y^β=z^γ with all of x,y,z of the same colour. Here c_r,α,β,γ>0 is some constant depending on the number of colours and the exponents in the equation. This is already a new result for α=β=1 and γ=2, that is to say for the equation x+y=z².

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