Prime power conjecture

Statement
The conjecture has the following equivalent formulations:


 * 1) Given a finite fact about::projective plane (viz., a projective plane with only finitely many points), its order (which is defined as one less than the number of points on any line) must be a fact about::prime power.
 * 2) The set of nontrivial prime powers (i.e., prime powers other than the number $$1$$) is precisely the set of numbers that can be realized as the orders of finite projective planes.
 * 3) Given a finite fact about::affine plane (viz., an affine plane with only finitely many points), its order (which is defined as the number of points on any line) must be a prime power.
 * 4) The set of nontrivial prime powers (i.e., prime powers other than the number $$1$$) is precisely the set of numbers that can be realized as the orders of finite affine planes.

Note that by the definition of projective plane and affine plane, the order of a projective plane or an affine plane cannot be $$1$$, hence we exclude $$1$$ from the set of possible orders when interpreting these statements.

Partial truth
It is known that every nontrivial prime power occurs as the order of a finite projective plane, so the question is whether for every number that is not a prime power, there is no projective plane of that order.

For specific numbers
The result is known for the case $$n = 6$$. In other words, there is no projective plane of order six. (Status for $$n = 10$$ and $$n = 12$$?)

General results
The following general results are true:


 * Bruck-Ryser theorem
 * -- with suitable transitivity assumptions, only have prime powers