Approximations of irrationalities by using linear recurrent sequences

The book is devoted to the approximation and periodic representation of algebraic numbers, like quadratic and cubic irrationalities. Continued fractions surely provide rational approximations and periodic representation for quadratic irrationals. Here, new kinds of representation for square roots are exploited with the aid of linear recurrent sequences. A beautiful connection with the solution of the Pell equation is highlighted, finding an original way to solve it through a group structure over the Pell hyperbola. Moreover, similar results are obtained and illustrated for cubic irrationals, approaching the Hermite problem, i.e., the problem of finding a way to write algebraic numbers (irrational numbers that are solution of some algebraic equation with rational coefficients) as a periodic sequence of integers.

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In 2007, I graduated in mathematics with honors. In 2011, I have obtained a PhD in mathematics with a thesis in the beautiful field of the number theory. Presently, I still do research in this field at the University of Turin and CNR of Pisa. As Gauss said: "Mathematics is the queen of sciences and number theory is the queen of mathematics".