>Prime powers plus projective planes

Prime powers plus projective planes

Mathematics is said to have infinitely more questions than answers. Here we present an open and famous problem of geometry called the "Prime Power Conjecture". A projective plane is a geometry consisting of points and lines such that

any pair of distinct lines have at least one point in common
there is a unique line through any pair of distinct points
any line has at least three points on it
there are at least two lines

Here are some examples of projective planes:

These planes have 7, 13, and 21 points respectively, and the number of lines is equal to the number of points. In fact, it is always true that for any projective plane, the number of lines is the same as the number of points, and there is a number n such that there are n²+n+1 points. We call n the order of the projective plane. So the projective planes above have orders 2, 3, and 4 respectively. In fact, it was proved by Lam, that there is no projective plane of order 10!

Nor is there a projective plane of order x where x-1 or x-2 is divisible by 4 and x is not the the sum of two squares (this is known as the Bruck-Ryser Theorem). Sounds complicated, but note that 6 is not allowed as 6-2 is 4. We can also rule out the following numbers 9, 14, 22, and so on. However, the following is a well-known fact:

What we know to be true:

Let p be a prime number (that is, a number which has exactly two divisors) and let m be any positive number. Then there exists a projective plane of order p^m.

So there are projective planes of orders 2,3,4,5,7,8,9,11,... but it is not known whether there exists a projective plane of order 12! Perhaps the following is true?

What we think could be true:

The order of any projective plane is a power of a prime.

This is known as the "Prime Power Conjecture", one of the most famous open problems in geometry. If it were true, then there would be no projective plane of order 12, since 12 is not a power of a prime.

Question: If we believe the Prime Power Conjecture, then why should a geometry of points and lines have to do with the prime numbers?

IMPORTANT IDEA
Mathematics is a DYNAMIC discipline with many interesting open problems yet to be solved. It is not a static `magic box-of-tricks' complete with answers to all questions.

Back Next Home