Math Conjectures

Existence of a Finite Projective Plane of Order 1212

OPENMajorCanonical finite caseProposed Unknown · Canonical special case

Canonical statement

Do there exist sets PP (points) and L\mathcal L (lines), each of size 122+12+1=15712^2+12+1=157, and an incidence relation I⊆P×LI\subseteq P\times\mathcal L, such that every line is incident with exactly 1313 points, every point is incident with exactly 1313 lines, every two distinct points lie on exactly one common line, and every two distinct lines meet in exactly one point?
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Do there exist sets \(P\) (points) and \(\mathcal L\)
(lines), each of size \(12^2+12+1=157\), and an incidence relation
\(I\subseteq P\times\mathcal L\), such that every line is incident with
exactly \(13\) points, every point is incident with exactly \(13\) lines,
every two distinct points lie on exactly one common line, and every two
distinct lines meet in exactly one point?
COMB-012:[Lam1991Plane10][Dembowski1968FiniteGeometries]
CombinatoricsDiscrete geometry

Notes

A projective plane of order 1212 would consist of 122+12+1=15712^2+12+1=157 points and 157157 lines, with 1313 points on every line and 1313 lines through every point, any two distinct points lying on exactly one common line and any two distinct lines meeting in exactly one point. The question of which orders admit projective planes is classical, with no single proposer or date; the general theory is laid out in Dembowski's treatise [Dembowski1968FiniteGeometries].

Every prime-power order qq is realized by the field construction, and no plane of non-prime-power order is known. The Bruck–Ryser–Chowla theorem excludes infinitely many candidate orders, but 1212 passes this test. The nearest resolved case is order 1010, ruled out by the massive computer-assisted search of Lam and collaborators [Lam1991Plane10], which makes 1212 the first order whose status is undecided.

Neither a construction nor a nonexistence proof for order 1212 is available, and a computational attack would be far larger than the order-1010 search; the case stands as the canonical open instance of the finite projective plane existence problem.

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.