Pell's equation
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Pell's equation is any Diophantine equation of the form
- <math>x^2-ny^2=1<math>
where n is a nonsquare integer. In calling it "Diophantine" we are really saying what we intend to do with the equation rather than describing any intrinsic property of the equation: we intend to seek solutions in which both x and y are integers. Infinitely many such solutions of this equation exist. The solutions yield good rational approximations to the square root of the natural number n.
The name of this equation arose from Leonhard Euler's mistakenly attributing its study to John Pell.
As motivation, consider the square root of two. It is often approximated 1.414..., which some might incorrectly interpret as 1.41414141414..., or 140/99. Likewise, the reciprocal of the square root of two to three decimal places is 0.707, which is suggestive of 0.70707070..., or 70/99. If 70/99 approximates the reciprocal of the square root of two, it follows that 99/70 approximates the square root of two. As it turns out, the square root of two is between 140/99 and 99/70. The arithmetic mean of these two rationals is 19601/13860. That number squared is 384199201/192099600. It turns out that 2 times the denominator 192099600 is 384199200, which differs from the numerator by only one. p = 19601 and q = 13860 satisfies the Diophantine equation 2q2 + 1 = p2. Any fraction of natural numbers p and q that satisfy this equation will be a reasonably good approximation for the square root of two.
More generally, if n is a given natural number, then any fraction of natural numbers p and q that satisfy the Pell's equation
- nq2 + 1 = p2
is a reasonably good approximation for the square root of n. The larger the numbers p and q, the better the approximation.
It turns out that if both (a, b) and (c, d) satisfy a Pell's equation, then so do
- <math>(bc + ad, bd + nac)<math>
and
- <math>(bc - ad, bd - nac).<math>
Fermat proved that p and q can always be found to satisfy a Pell's equation for any natural number n that is not a perfect square. Given a computer with bignum capability, this makes it easy to converge rapidly toward any irrational square root of a n. As an added bonus, a Pell's equation can always be solved in a finite number of steps by calculating the continued fraction representation of the square root of n.
