Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T00:58:50.418Z Has data issue: false hasContentIssue false

Non-existence of a certain projective plane

Published online by Cambridge University Press:  09 April 2009

R. H. F. Denniston
Affiliation:
University of Leicester
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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