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Factorization and congruence in the arithmetics of Cayley's algebra

Published online by Cambridge University Press:  18 May 2009

P. J. C. Lamont
P. J. C. Lamont
Affiliation:
Department of Computer Science, College of Applied Sciences, Western Illinois University, Macomb, Illinois 61455
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This paper treats factorization and congruence in the arithmetics of Cayley's algebra C. Results, proved by Rankin [10], on the number of factorizations of a given element in the nonmaximal arithmetics Ho and Jo are reviewed. Further, new results on congruence are established and are used to find the number of factorizations of a prescribed element in the remaining arithmetics, including the maximal arithmetics Jw When computer factorizing a given element, the congruence results can be used as a heuristic device to prune the search tree.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 2 , May 1991 , pp. 171 - 180
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.van der Blij, F., History of the octaves, Wis- en Natuurkundig Tijdschrift (= Simon Stevin) 34, III (1961), 106–125.Google Scholar
2.Van der Blij, F. and Springer, T. A., The arithmetics of the octaves and of the group G2, Nederl. Akad. Wetensch. Proc. (=Indag. Math.) 62A (1959), 406–418.Google Scholar
3.Lamont, P. J. C., Arithmetics in Cayley's algebra, Proc. Glasgow Mathematical Assoc. 6 (1963), 99–106.Google Scholar
4.Lamont, P. J. C., Ideals in Cayley's algebra, Nederl. Akad. Wetensch. Proc. (=Indag. Math.) 66A (1963), 394–400.Google Scholar
5.Lamont, P. J. C., Approximation theorems for the group G2, Nederl. Akad. Wetensch. Proc. (=Indag. Math.) 67A (1964), 187–192.Google Scholar
6.Lamont, P. J. C., Factorization and arithmetic functions for orders in composition algebras, Glasgow Mathematical Journal 14 (1973), 86–95.Google Scholar
7.Lamont, P. J. C., The number of Cayley integers of given norm, Proc. Edinburgh Math. Soc. 25 (1982), 101–103.Google Scholar
8.Lamont, P. J. C.. Computer generated natural inner automorphisms of Cayley's algebra, Glasgow Mathematical Journal 23 (1982), 187–189.Google Scholar
9.Lamont, P. J. C., The nonexistence of a factorization formula for Cayley numbers, Glasgow Mathematical Journal 24 (1983), 131–132.Google Scholar
10.Rankin, R. A., A certain class of multipicative functions, Duke Math. J. 13 (1946), 281–306.Google Scholar
11.Taussky, O., Sums of squares, Am. Math. Monthly 77 (1970), 805–830.Google Scholar