6
$\begingroup$

Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive convergents of its simple continued fraction, where $k\ge 1$.

Is it possible that both $$|a-(p_k/q_k)|<1/(2q_k^2)$$ and $$|a-(p_{k+1}/q_{k+1})|<1/(2q_{k+1}^2)$$ are true?

I can only prove that at least one of these inequalities is true.

  • 0
    Do you have $\lfloor a \rfloor = p_0/q_0$?2011-03-09
  • 0
    Yes. That is right.2011-03-09

2 Answers 2