Editing Pell's equation (section)

===Chebyshev polynomials===
Demeyer mentions a connection between Pell's equation and the [[Chebyshev polynomials]]:
If <math>T_i(x)</math> and <math>U_i(x)</math> are the Chebyshev polynomials of the first and second kind respectively, then these polynomials satisfy a form of Pell's equation in any [[polynomial ring]] <math>R[x]</math>, with <math>n = x^2 - 1</math>:<ref>{{Citation |last=Demeyer |first=Jeroen |title=Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields |url=http://cage.ugent.be/~jdemeyer/phd.pdf |page=70 |year=2007 |archive-url=https://web.archive.org/web/20070702185523/https://cage.ugent.be/~jdemeyer/phd.pdf |series=PhD thesis, [[Ghent University]] |access-date=27 February 2009 |archive-date=2 July 2007 |url-status=dead}}.</ref>
<math display="block">T_i^2 - (x^2-1) U_{i-1}^2 = 1.</math>
Thus, these polynomials can be generated by the standard technique for Pell's equations of taking powers of a fundamental solution:
<math display="block">T_i + U_{i-1} \sqrt{x^2-1} = (x + \sqrt{x^2-1})^i.</math>
It may further be observed that if <math>(x_i, y_i)</math> are the solutions to any integer Pell's equation, then <math>x_i = T_i (x_1)</math> and <math>y_i = y_1 U_{i-1} (x_1)</math>.<ref>{{Citation |last=Barbeau |first=Edward J. |title=Pell's Equation |url=https://archive.org/details/pellsequation0000barb |chapter=3. Quadratic surds |year=2003 |series=Problem Books in Mathematics |publisher=Springer-Verlag |isbn=0-387-95529-1 |mr=1949691 |url-access=registration}}.</ref>