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 2*q*^{2} + 1 = *p*^{2}. 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 **Pell's equation**

*nq*^{2}+ 1 =*p*^{2}

It turns out that if both and satisfy Pell's equation, then so do