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Green and Lindqvist proved that for any 2‐colouring of   N  , there are infinitely many monochromatic solutions to    x  +  y  =   z  2    . In fact, they showed the existence of a monochromatic solution in every interval    [  N  ,  c   N  8   ]    with large enough   N  . In this short note we give a different proof for their theorem and prove that a monochromatic solution exists in every interval    [  N  ,   10  4    N  4   ]    with large enough   N  . A 2‐colouring of    [  N  ,   (  1  /  27  )    N  4   ]    avoiding monochromatic solutions to    x  +  y  =   z  2is also presented, which shows that in10  4    N  4only the constant factor can be reduced.

Monochromatic solutions to x + y = z 2 in the interval [ N , c N 4 ]