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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