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### Subject: sci.math FAQ: Projective Plane of Order 10

This article was archived around: Fri, 17 Nov 1995 17:15:47 GMT

Archive-Name: sci-math-faq/proyectiveplane
Last-modified: December 8, 1994
Version: 6.2

PROJECTIVE PLANE OF ORDER 10
More precisely:
Is it possible to define 111 sets (lines) of 11 points each such that:
For any pair of points there is precisely one line containing them
both and for any pair of lines there is only one point common to them
both?
Analogous questions with n^2 + n + 1 and n + 1 instead of 111 and 11
have been positively answered only in case n is a prime power. For n =
6 it is not possible, more generally if n is congruent to 1 or 2 mod 4
and can not be written as a sum of two squares, then an FPP of order n
does not exist. The n = 10 case has been settled as not possible
either by Clement Lam. As the ``proof" took several years of computer
search (the equivalent of 2000 hours on a Cray-1) it can be called the
most time-intensive computer assisted single proof. The final steps
were ready in January 1989.
References
R. H. Bruck and H. J. Ryser. The nonexistence of certain finite
projective planes. Canadian Journal of Mathematics, vol. 1 (1949), pp
88-93.
C. Lam. American Mathematical Monthly, 98 (1991), 305-318.
_________________________________________________________________
alopez-o@barrow.uwaterloo.ca
Tue Apr 04 17:26:57 EDT 1995