# Pell's Equation

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## Pell’s Equation

an equation of the form *x*^{2} - *Dy*^{2} = 1, where *D* is a positive integer that is not a perfect square and the equation is to be solved in integers. The equation has an infinite number of solutions. The solution *x*_{0} = 1, *y*_{0} = 0 is obvious. The next—in terms of magnitude—solution (*x*_{1}, *y*_{1}) of Pell’s equation can be found by expanding into a continued fraction. If we know the solution (*x*_{1}, *y*_{1}), the entire set of solutions (*x _{n} y_{n}*) can be obtained by using the formula

Pell’s equation is closely related to the theory of algebraic numbers. It is named after the 17th-century British mathematician J. Pell, to whom L. Euler incorrectly attributed a method of solving this equation.

### REFERENCES

Venkov, B. A.*Elementarnaia teoriia chisel*. Moscow-Leningrad, 1937. Chapter 2.

Dickson, L. E.

*History of the Theory of Numbers*, vol. 2. New York, 1966.