The centrifuge problem asks which integers 
allow a selection of 
distinct 
th roots of unity whose sum is zero, where 
is an integer and 
. It models placing identical masses in 
equally spaced positions around a circle,
with at most one mass in each position (Sivek 2010, Baker 2018, Numberphile 2018).
For a subset 
of indices 0, 1, ..., 
with 
, the condition is
 |
The empty set is allowed when 
.
Let 
,
..., 
be the distinct prime factors of 
, and define
 |
A suitable selection exists iff both 
and 
(Sivek 2010). The second condition reflects the fact
that the unselected roots of unity also have sum zero. For example, when 
, the choice 
fails despite 
because 
cannot be expressed as such a sum.
If repeated roots of unity are allowed instead, only
is required (Lam and Leung 2000).
For a prime power 
, the condition reduces to 
.
