Pell's Equation

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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 Pells Equation into a continued fraction. If we know the solution (x1, y1), the entire set of solutions (xn yn) 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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Cohen had written a computer program specifically to look for what mathematicians call Fermat "near misses": combinations of numbers a, b, c, and n that come so close to satisfying Fermat's equation that they would seem to work when tested on a calculator.
As one consequence of his proof of the Mordell conjecture (SN: 7/23/83, p.58), he showed that if there are any solutions to Fermat's equations, then there are only a finite number of them for each value of n.
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