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 x2 + y2 = z2. He proved that if x and y are relatively prime and x − y is positive and odd, then (x, y, z) = (x2 − y2, 2xy, x2 + y2) is a Pythagorean triple x2 + y2 = z2, and conversely all primitive Pythagorean triples arise in this manner.
A standard reference on diophantine approximation is Cassels .
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 ...|