Math Conjectures

Prime-Power Conjecture for Finite Projective Planes

OPENLandmarkConjectureProposed Unknown · Full conjecture

Canonical statement

Every finite projective plane has prime-power order. Equivalently, if a finite projective plane has q+1q+1 points on every line and q+1q+1 lines through every point, then qq is a power of a prime.
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Every finite projective plane has prime-power order. Equivalently, if a finite projective plane has \(q+1\) points on every line and \(q+1\) lines through every point, then \(q\) is a power of a prime.
COMB-DESIGN-017:[BruckRyser1949Nonexistence][Dembowski1968FiniteGeometries][Lam1991Plane10]
CombinatoricsDiscrete geometry

Notes

A finite projective plane of order n has n²+n+1 points and the same number of lines, with n+1 points on every line and a unique line through each pair of points. Finite fields construct such a plane whenever n is a prime power [Dembowski1968FiniteGeometries]. The prime-power conjecture asserts that these are the only possible orders.

The Bruck–Ryser–Chowla obstruction rules out many non-prime-power orders by arithmetic conditions, but gives only a necessary condition [BruckRyser1949Nonexistence]. A massive computer-assisted argument proved that no projective plane of order 10 exists [Lam1991Plane10]. Order 12 is the smallest undecided order, yet resolving that single case would not settle the umbrella conjecture: what remains is either a non-prime-power construction or a theorem excluding every such order.

References (3)

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