Pell's Equation

Pell’s Equation

an equation of the form x² - Dy² = 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 x₀ = 1, y₀ = 0 is obvious. The next—in terms of magnitude—solution (x₁, y₁) of Pell’s equation can be found by expanding into a continued fraction. If we know the solution (x₁, y₁), 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.

单词	pell's equation
释义	Pell's Equation Pell’s Equation an equation of the form x² - Dy² = 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 x₀ = 1, y₀ = 0 is obvious. The next—in terms of magnitude—solution (x₁, y₁) of Pell’s equation can be found by expanding into a continued fraction. If we know the solution (x₁, y₁), 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.
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