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In modular arithmetic, Thue's lemma roughly states that every modular integer may be represented by a "modular fraction" such that the numerator and the denominator have absolute values not greater than the square root of the modulus.
More precisely, for every pair of integers with m > 1, given two positive integers X and Y such that X ≤ m < XY, there are two integers x and y such that
and
Usually, one takes X and Y equal to the smallest integer greater than the square root of m, but the general form is sometimes useful, and makes the uniqueness theorem easier to state.
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