Continuing in the projective geometry vain --

If N-1 is prime, one can construct a projectve plane with N points on each line as follows:

The points are:

1. Ordered pairs (x,y) where x and y are each integers in the range 0 through N-2 inclusive.

2. "Ordered singles" m in the range 0 through N-2 inclusive.

3. An "ordered zero" (an extra phantom point with no particular name).

The lines are:

A. Equations of the form y=xM+B where M and B are integers in the range 0 through N-2 inclusive.

B. Equations of the form x=A where A is an integer in the range 0 through N-2 inclusive.

C. An extra phantom line with no equations attached.

Points of type 1 are called "ordinary points". Points of types 2 and 3 are called "points at infinity".

Lines of types A and B are called "ordinary lines". The line of type C is called the "line at infinity".

An ordinary point is defined to be on an ordinary line in case the coordinates satisfy the equation. (All arithmetic is done modulo N-1). A type 2 point at infinity is defined to be on a type A ordinary line in case m=M. The type 3 point at infinity is on all type B ordinary lines. All points at infinity (types 2 and 3) are on the line at infinity (type C). No other "on" relationships exist, e.g. no ordinary point is on the line at infinity.

All of the above works only if N-1 is prime. Otherwise, the arithmetic and geometry don't work, e.g. if N=5 we have 2*2=0 which really screws things up.

That's not to say projective planes don't exist with N=5, just that they can't be constructed as described above. Such planes probably wouldn't enjoy the Pappas and Desargues properties (for those of you who know about such things), however.

Bill Smythe