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Gregory Galperin invented billiard method of computing $\pi$, see Playing Pool With $\pi$ (The Number $\pi$ From A Billiard Point Of View) To calculate $\pi$, take two identical balls. Put one near a …

Jürgen Moser used finite contined fractions (functional Stieltjes contined fractions) in his solution of the open Toda lattice

Journal of Fixed Point Theory and Applications. V.3, No.1, 2008, 79–93. …

Nearest integer continued fractions (NICF) can be applied to analysis of number's spectra and "eta-sequences". The spectrum of a real number is defined to be an infinite multiset of integers (see "Con …

M. Skopenkov gave a reference to criterion for a rectangle to be tilable by rectangles of a similar shape: C. Freiling, D. Rinne, Tiling a square with similar rectangles, Math. Res. Lett. 1 (1994) 547 …

[Lidl, R. & Niederreiter, H. Finite fields, pp. 237-239.] For $q$ odd, put $G(x) = x^q - x$, let $f\in\mathbb{F}_q[x]$ be a polynomial of positive degree with no roots in $\mathbb{F}_q$, and set $F(x …

In the article Factoring large integers Lehman proposed a modification of Fermat's difference of squares method for factoring large integers. This modification permits factoring $n$ in $ O({n^{1/3}})$ …

Let $(a,b)=1$, $\pi(x)=ax\pmod b$ and $I(a,b)$ denote the number of inversions of permutation $$(0,1,\ldots, b-1)\to(\pi(0),\pi(1),\ldots,\pi (b-1)).$$ Zolotarev proved quadratic reciprocity law using …

Markov chains have applications in speech recognition (see A tutorial on hidden Markov models and selected applications in speech recognition by Rabiner), analysis of DNA sequences (see Biological Sequence … Analysis: Probabilistic Models of Proteins and Nucleic Acids by Durbin, Eddy, Krogh, Mitchison), in Google PageRank engine etc, see Wikipedia article for a longer list of applications. …

Also there are some examples with visible and nonvisible lattice points: Fritz Herzog and B. M. Stewart, Patterns of Visible and Nonvisible Lattice Points, The American Mathematical Monthly, Vol. 78 …

I know some applications of finite continued fractions. Probably you know more. Can you add anything? … Odlyzko (eds.), Emerging Applications of Number Theory, IMA Volumes in Mathematics and its Applications, Volume 109 (Springer, New York, 1999) pp. 405-450) 19) The statistics of the trajectory of Sinai …

Cornacchia's algorithm for solving the Diophantine equation $x^{2}+dy^{2}=m$. See for details On Cornacchia’s algorithm for solving the diophantine equation $x^{2}+dy^{2}=m$ by F. Morain and J.-L. Nic …