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Non-existence of a certain projective plane

Published online by Cambridge University Press:  09 April 2009

R. H. F. Denniston
Affiliation:
University of Leicester
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The question whether any projective plane of order ten exists or not, is an unsolved problem that has attracted some interest (see, for instance, [2]). A method, by which a plane might have been discovered, was suggested to me by a theorem in [1]: ‘If order of a plane is greater than 10, a six-arc is not complete’. Elementary arguments do not, it appears, exclude the possibility of a complete six-arc in a plane of order ten: but they do show that such a figure would be of an extreme type, and that the whole plane would fit round it in a particular way. The limitation, in fact, is so severe that it becomes feasible to consider, for a good many of the incidences in the plane, all the alternative arrangements that seem possible. With the help of the Elliott 4130 computer of the University of Leicester, I have carried out an exhaustive search, and discovered that it is impossible to build up a projective plane by this method. So I can assert:

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Martin, G. E., ‘On arcs in a finite projective plane’, Canadian J. Math. 19, (1967), 376393.CrossRefGoogle Scholar
[2]Parker, E. T., ‘Orthogonal latin squares’, Proc. Nat. Acad. Sci. U. S. A. 45, (1959), 859862.CrossRefGoogle ScholarPubMed