Suppose we are given a rational q. Is it possible that there infinitely many integer solutions ( h , k ) to following system of inequalities: 0 &lt; | q − h / k | &lt; 1 / 2 k 2 ? I think that it is easy to see that for q = 1 / 2 (and probably for any rational a / b with 2 | b and g c d ( a , b ) = 1) it is impossible, but what about other rationals? What about changing constant in the right inequality (making it stronger or weaker)?

Question

Suppose we are given a rational   q. Is it possible that there infinitely many integer solutions   (  h  ,  k  ) to following system of inequalities:   0  &amp;lt;  |  q  −  h  /  k  |  &amp;lt;  1  /  2      k    2  ? I think that it is easy to see that for   q  =  1  /  2 (and probably for any rational   a  /  b with   2  |  b and   g  c  d  (  a  ,  b  )  =  1) it is impossible, but what about other rationals? What about changing constant in the right inequality (making it stronger or weaker)?

vrtuljakwb · Accepted Answer

Suppose   q  =      a    b   is rational, and       h    k    ≠  q. Then we have        |        a    b    −      h    k          |    =        |              a      k      −      b      h              b      k              |    ⩾      1          |      b      k      |        .Thus, for every   c  &amp;gt;  0 there can be only finitely many pairs   (  h  ,  k  ) such that  0  &amp;lt;        |    q  −      h    k          |    &amp;lt;      c          k      2        .

pipantasi4 · Accepted Answer

I had a problem with this question, thanks for the solution!

Suppose we are given a rational q . Is it possible that there infinitely many integer solutions

Answered question

Answer & Explanation

New Questions in Algebra I