Michele Elia
Published 2019-05-26Version 1
Legendre found that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. Similarly, it is shown here that the continued fraction expansion of $\sqrt N$ having even period directly produces a factor of a composite $N$. Shanks' infrastructural method is then revisited, and some consequences of its application to factorization by means of the continued fraction expansion of $\sqrt N$ are derived.
On the period of the continued fraction expansion of ${\sqrt {2^{2n+1}+1}}$
Limit theorems related to beta-expansion and continued fraction expansion
The first and second moment for the length of the period of the continued fraction expansion for $\sqrt{d}$
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