PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.)
Vol. 59(73), pp. 18--30 (1996)
Involutions associated with sums of two squares
P. ShiuDepartment of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, United Kingdom
Abstract: In 1984 D.R. Heath-Brown constructed two involutions from which a new and simple proof of Fermat's theorem on the decomposition of a prime $$p\equiv 1\pmod 4$ as a sum of two squares was derived. An algorithm based on the composition of the two involutions is constructed for the decomposition of $p$, and the method can also be used for the factorisations of suitable composite numbers. The process corresponds to the continued fraction expansion of a reduced quadratic irrational related to $\sqrt p$, and the period of the composite map is the sum of the relevant partial quotients.
Keywords: Fermat's two square theorem, involutions, periods factorisation, continued fractions
Classification (MSC2000): 11A51; 11Y05
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