It was Pierre de Fermat who brought Europe's math community to focus on this equation in 1657 when he posed the problem as a challenge to the British mathematicians. Fermat was not the first to identify the problem. Diophantus studied one variant of Pell's equation and a famous problem by Archimedes known as the Cattle Problem can be stated in terms of Pell's equation. Great progress in solving Pell's equation were made by Indian mathematicians by Brahmagupta (born in 598 AD) and Bhascara Acharya (born in 1114 AD).

Pell's equation amounts to this:

Find all integer solutions for x,y where Ax

^{2}+ 1 = y

^{2}.

Interestingly, when this problem got passed on to the English, the part about "integers" was dropped and the British mathematicians quickly found a solution for rational solutions.

Here's the solution in terms of rational values:

Lemma: There are an infinite number of rational solutions to Ax

^{2}+ 1 = y

^{2}where x,y are rational numbers.

(1) Let m = y-1, n = x. Then we know that: (y -1)/x = m/n and y-1 = (m/n)x and y = 1 + (m/n)x.

(2) So, Ax

^{2}+ 1 = [1 + (m/n)x]

^{2}= 1 + 2(m/n)x + (m

^{2}/n

^{2})x

^{2}

(3) Multiplying n

^{2}to both sides gives us:

An

^{2}x

^{2}+ n

^{2}= n

^{2}+ 2mnx + m

^{2}x

^{2}

And:

An

^{2}x

^{2}= 2mnx + m

^{2}x

^{2}

And:

An

^{2}x

^{2}- m

^{2}x

^{2}= 2mnx

(4) Dividing x from both sides gives us:

An

^{2}x - m

^{2}x = 2mn = x(An

^{2}- m

^{2})

So that:

x = 2mn/(An

^{2}- m

^{2})

And:

y = 1 + (m/n)x = 1 + (m/n)(2mn/[An

^{2}- m

^{2}]) =1 + 2m

^{2}/(An

^{2}- m

^{2}) = (An

^{2}- m

^{2}+ 2m

^{2})/(An

^{2}- m

^{2}) = (An

^{2}+ m

^{2})/(An

^{2}- m

^{2})

QED

When Fermat saw the solution for rational values, he rejected it explaining that it was ridiculous to think that he would offer a problem as simple as this. He clarified that the problem is only interesting when the solution is for integers.

The British mathematicians at first responded that Fermat's new problem was artificial. They saw little justification for solving the problem specifically for whole numbers when the solution for rational values was valid. Later, an integer solution and a method for finding solutions was put forward without any proof that the method worked in all circumstances. Historically, Lord Brouncker is given credit for the solution.

The verification of the British method would have to wait over 100 years. It was the great mathematician Joseph-Louis Lagrange, using continued fractions, who showed that the British method worked in all circumstances. Go here to see how continued fractions can be used to provide a solution to Pell's equation.

References

- Harold M. Edwards Fermat's Last Theorem
- Pell's Equation, Wikipedia
- Pell's Equation, MacTutor

## 2 comments:

In Step(3), last line:

An^2 - m^2x^2 = 2mnx

Should be:

An^2x^2 - m^2x^2 = 2mnx ?

Rob

Hi Rob,

Thanks for noticing. I fixed the typo.

Cheers,

-Larry

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