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number > number theory > algebraic number theory > modular arithmetic > Thue's lemma

number > real number > Diophantine approximation > Thue's lemma

Terme préférentiel

Thue's lemma

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 $(a, m)$ with $m > 1$ , given two positive integers $X$ and $Y$ such that $X \leq m < XY$ , there are two integers $x$ and $y$ such that
${\\displaystyle ay\\equiv x{\\pmod {m}}}$
and
${\\displaystyle |x|<X,\\quad 0<y<Y.}$
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 (below) easier to state.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Thue%27s_lemma)

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