[4]) In general, the solution whose existence is asserted by Thue's lemma is not unique.
Therefore, one may only hope for uniqueness for the rational number x/y, to which a is congruent modulo m if y and m are coprime.
[5] The proof is rather easy: by multiplying each congruence by the other yi and subtracting, one gets The hypotheses imply that each term has an absolute value lower than XY < m/2, and thus that the absolute value of their difference is lower than m. This implies that
The original proof of Thue's lemma is not efficient, in the sense that it does not provide any fast method for computing the solution.
[6] More precisely, given the two integers m and a appearing in Thue's lemma, the extended Euclidean algorithm computes three sequences of integers (ti), (xi) and (yi) such that where the xi are non-negative and strictly decreasing.