Exercise 6.1.6 (The condition $ad -bc = 1$ in problem 5 is not sufficient)

Question

Prove that the two fractions described in the preceding problem are not necessarily adjacent in the Farey sequence of order $n+1$.

Richard Ganaye · Accepted Answer

\begin{proof} 
A counterexample is sufficient: $3/4$ and $1/1$ are adjacent in the Farey sequence of order $4$, but not in the Farey sequence of order $5$, since
$$\frac{3}{4} < \frac{4}{5} < \frac{1}{1}.$$
(See table p. 298.)

Here $4\cdot 1 - 3 \cdot 1 = 1$, but $\max(1,4) < 5$, so the hypotheses of Problem 5 are not satisfied.

\end{proof}

Exercise 6.1.6 (The condition $ad -bc = 1$ in problem 5 is not sufficient)

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Exercise 6.1.6 (The condition $ad -bc = 1$ in problem 5 is not sufficient)

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