of two possible counterexamples must be a valid counterexample, without showing which one it is. Pythagorean proof of the existence of irrational numbers The proof by Pythagoras (or more likely one of his students) about 500 BCE has had a profound effect on mathematics. It shows that the square root of 2 cannot be expressed as the ratio of two integers. The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers. This bifurcation was used by Cantor in his diagonal method, which in turn was used by Turing in his proof that the Entscheidungsproblem, the decision problem of Hilbert, is undecidable. Proofs followed for various square roots of the primes up to 17. There is a famous passage in Plato's Theaetetus in which it is stated that Theodorus (Plato's teacher) proved the irrationality of taking all the separate cases up to the root of 17 square feet ... . A more general proof now exists that: The mth root of an integer N is irrational, unless N is the mth power of an integer n". That is, it is impossible to express the mth root of an integer N as the ratio of two integers a and b, that share no common prime factor except in cases in which b = 1. Impossible constructions sought by the ancient Greeks Three famous questions of Greek geometry were how: to trisect any angle using a compass and a straightedge, to construct a cube with a volume twice the volume of a given cube, to construct a square equal in area to that of a given circle. For more than 2,000 years unsuccessful attempts were made to solve these problems; at last, in the 19th century it was proved that the desired constructions are logically impossible. A fourth problem of the ancient Greeks was to construct an equilateral polygon with a specified number n of sides, beyond the basic cases n = 3, 4, 5, 6 that they knew how to construct. All of these are problems in Euclidean construction, and Euclidean constructions can be done only if they involve only Euclidean numbers (by definition of the latter). Irrational numbers can be Euclidean. A good example is the square root of 2 (an irrational number). It is simply the length of the hypotenuse of a right triangle with legs both one unit in length, and it can be