Diophantus was a Greek geometer who developed the theory of equations with integer solutions, a subject now referred to as *diophantine equations*.^{1} Diophantus determined all the integer *Pythagorean triples* (*x, y, z*), solutions of *x*^{2} + *y*^{2} = *z*^{2}. He proved that if *x* and *y* are relatively prime and *x* − *y* is positive and odd, then (*x, y, z*) = (*x*^{2} − *y*^{2}, 2*xy*, *x*^{2} + *y*^{2}) is a Pythagorean triple *x*^{2} + *y*^{2} = *z*^{2}, and conversely all primitive Pythagorean triples arise in this manner.

A standard reference on diophantine approximation is Cassels [1957].

*Diophantine approximation* studies the accuracy with which a real number *x* can be approximated by a rational number *p/q*. The accuracy of the approximation is measured by ||*x − p/q*||, where

It should be obvious that an approximation by rational numbers *p/q* of a real number, say *π* = 3.1415927…, is improved by increasing *q*. A basic result is

**Proposition 11.10:** [Cassels, 1957]:

11.10a |
Given x and Q > 1, there exists an integer q with 0 < q < Q such that ||qx|| ≤ Q^{−1}. |

11.10b |
There are infinitely many integers q such that ||qx|| < q^{−1}. |

11.10c |
For every ∈ > 0 and real number x there are only finitely many integers q such that ||qx|| < q^{− 1 − ∈}. |

11.10d |
If ||qx|| < 1, there exists an integer p such that ||qx|| = |qx − p| < 1. Equivalently, |z − p/q| < 1, which asserts that p is the best choice for the numerator for the ... |

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