= Zero-residue constraint for idempotent ultrafilters
{title2=$p\text{ selects }0\pmod q$}
An <ultrafilter> on the positive integers selects exactly one <residue class> modulo a fixed positive integer $q$. Under <addition on the Stone-Čech compactification of the natural numbers>, selected residues add. An <idempotent ultrafilter> must therefore select a residue $r$ with $2r=r$ in the finite cyclic group, hence $r=0$. In particular, the multiples of every fixed positive integer belong to every additive <idempotent ultrafilter>.
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