The problem of finding integer relations among a set of real numbers
was first studied by Euclid, who gave an iterative algorithm (the
Euclidean algorithm), which when applied to two real numbers, either
terminates, yielding an exact relation, or produces an infinite
sequence of approximate relations. The generalization of this problem
for n > 2 has been attempted by Euler, Jacobi, Poincare, Minkowski,
Perron, Brun, Bernstein, among others. However, none of their
algorithms has been proven to work for n > 3, and numerous
counterexamples have been found.
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