The Non-Existence of Finite Projective Planes of Order 10

A finite projective plane of order n, with n > 0, is a collection of n2+ n + 1 lines and n2+ n + 1 points such that 1. every line contains n + 1 points, 2. every point is on n + 1 lines, 3. any two distinct lines intersect at exactly one point, and 4. any two distinct points lie on exactly one line. It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.