So far, I have put up the following posts related to finite projective planes: Primitive roots Finite fields Finite fields (2) Latin squares, solutions, more solns Desargues theorem Projective planes, solutions Pappus and Desargues Bruck-Ryser (1) Bruck-Ryser (2) The objective is to give the apparently relevant background to the long-standing conjecture that there are no [...]
April was certainly mensis horribilis for Cameron. It ended with an unfortunate exchange on Monday: Mr Dennis Skinner (Bolsover) (Lab) Why is the Culture Secretary getting better employment rights than the rest of the workers in Britain? Is it possibly because the Prime Minister knows that as long as the Culture Secretary is in the [...]
In the last post we showed that: If \(n\) is 1 or 2 mod 4 and there is a finite projective plane of order \(n\), then \(n\) can be written as the sum of two squares of rational numbers. The objective now is to show that if a prime \(p=3\bmod 4\) divides \(n\) to an [...]
Our objective is to prove the following result: Bruck-Ryser theorem If \(n\) is 1 or \(2\bmod 4\) and has a prime factor \(p=3\bmod 4\) which divides it to an odd power, then there is no finite projective plane of order \(n\). [\(p\) divides \(n\) to an odd power has the obvious meaning that the highest [...]
In the previous post on Desargues’ theorem we showed the classic diagram which is how it looks in Euclidean geometry in the general case where the lines are not parallel. But this is not really a fair representation of the general projective situation, which is much more symmetrical than that implies. The best description of [...]
