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For the case ''n'' = 2, an extension of the [[Euclidean algorithm]] can determine whether there is an integer relation between any two real numbers ''x''<sub>1</sub> and ''x''<sub>2</sub>. The algorithm generates successive terms of the [[continued fraction]] expansion of ''x''<sub>1</sub>/''x''<sub>2</sub>; if there is an integer relation between the numbers then their ratio is rational and the algorithm eventually terminates.
The first general algorithm that was proved to work for all values of ''n'' was the [[Ferguson–Forcade algorithm]], published in 1979 by [[Helaman Ferguson]] and [[R.W. Forcade]].<ref>{{MathWorld|urlname=IntegerRelation|title=Integer Relation}}</ref> Subsequent developments, focussing on improving both efficiency and numerical stability, produced the following algorithms:
*The [[Lenstra–Lenstra–Lovász lattice basis reduction algorithm|'''LLL algorithm''']], developed by [[Arjen Lenstra]], [[Hendrik Lenstra]] and [[László Lovász]] in 1982.<ref>{{MathWorld|urlname=LLLAlgorithm|title=LLL Algorithm}}</ref>
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