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The Practical Calculation of Unit Algebraical Numbers

Published online by Cambridge University Press:  24 October 2008

H. C. Pocklington
H. C. Pocklington
Affiliation:
St John's College
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Extract

The object of this paper is to give a method of finding the units of a set of algebraical numbers that shall be convenient practically. Little attention is given to theoretical considerations. The case of numbers composed with the cube root of an integer is treated in § 2, and a few theoretical remarks are added in § 3. In § 4 we show how to test whether the unit found is a fundamental one or not. In § 5 the method is extended to other cases.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 24 , Issue 4 , October 1928 , pp. 471 - 476
Copyright
Copyright © Cambridge Philosophical Society 1928

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References

* It is nearly as good to make the coefficients of a as small as possible, as we do in most of the cases of §5.Google Scholar

† The co-factor of x + a y + a 2z is (x 2−Dyz)+a(Dz 2−xy)+a 2(y 2−xz).Google Scholar

* Todd, H., Proc. Camb. Phil. Soc., Vol. xix, p. 111 (1917).Google Scholar

† This method is sometimes inapplicable. E.g. for D = 10 the fundamental unit is E = 1 + 6a − 3a 2 where 9(1 + 6a − 3a 2) = (7 + a − 2a 2)2 and E is a quadratic residue of every prime. Here we equate E or 1/E to the square of x + ay + a 2z and exclude by moduli or try all numbers between certain limits.Google Scholar