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The Non-Existence of Finite Projective Planes of Order 10

Published online by Cambridge University Press:  20 November 2018

C. W. H. Lam ,
L. Thiel  and
S. Swiercz
C. W. H. Lam
Affiliation:
Concordia University, Montréal, Québec
L. Thiel
Affiliation:
Concordia University, Montréal, Québec
S. Swiercz
Affiliation:
Concordia University, Montréal, Québec
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A finite projective plane of order n, with n > 0, is a collection of n2+ n + 1 lines and n2+ n + 1 points such that

1. every line contains n + 1 points,

2. every point is on n + 1 lines,

3. any two distinct lines intersect at exactly one point, and

4. any two distinct points lie on exactly one line.

It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 6 , 01 December 1989 , pp. 1117 - 1123
Copyright
Copyright © Canadian Mathematical Society 1989

References

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