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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

All FAQs in Directory: sci-math-faq
All FAQs posted in: sci.math
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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