= Leading-residue obstruction to partition regularity
{title2=$c(x)=p^{-v_p(x)}x\bmod p$}
For nonzero <integer> coefficients $a_i$, choose a <prime number> $p>\sum_i|a_i|$ and colour each positive integer by its nonzero leading residue after removing its power of $p$. In a <monochromatic> zero-sum solution, divide by the least occurring power of $p$. Reduction modulo $p$ then makes the sum of the coefficients attached to the least-valuation variables vanish modulo $p$. Its absolute value is less than $p$, so that sum vanishes as an integer. This gives the necessity direction of the one-equation <partition regularity> criterion without using the general <Rado theorem>.
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