6829, 8269, 8941, 9949, ... . (For these records, see for x and for y.) List of fundamental solutions of Pell's equations The following is a list of the fundamental solution to with n ≤ 128. When n is an integer square, there is no solution except for the trivial solution (1, 0). The values of x are sequence and those of y are sequence in OEIS. Connections Pell's equation has connections to several other important subjects in mathematics. Algebraic number theory Pell's equation is closely related to the theory of algebraic numbers, as the formula is the norm for the ring and for the closely related quadratic field . Thus, a pair of integers solves Pell's equation if and only if is a unit with norm 1 in . Dirichlet's unit theorem, that all units of can be expressed as powers of a single fundamental unit (and multiplication by a sign), is an algebraic restatement of the fact that all solutions to the Pell's equation can be generated from the fundamental solution. The fundamental unit can in general be found by solving a Pell-like equation but it does not always correspond directly to the fundamental solution of Pell's equation itself, because the fundamental unit may have norm −1 rather than 1 and its coefficients may be half integers rather than integers. Chebyshev polynomials Demeyer mentions a connection between Pell's equation and the Chebyshev polynomials: If and 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 , with : Thus, these polynomials can be generated by the standard technique for Pell's equations of taking powers of a fundamental solution: It may further be observed that if are the solutions to any integer Pell's equation, then and . Continued fractions A general development of solutions of Pell's equation in terms of continued fractions of can be presented, as the solutions x and y are approximates to the square root of n and thus are a special case of continued fraction approximations for quadratic irrationals. The relationship to the continued fractions implies that the solutions to Pell's equation form a semigroup subset of the modular group. Thus, for example, if p and q satisfy Pell's equation, then is a matrix of unit determinant. Products of such matrices take exactly the same form, and thus all such products