Pierce expansions


A discrete iteration related to Pierce expansions ★★

Author(s): Shallit

\begin{conjecture} Let $a > b > 0$ be integers. Set $b_1 = b$ and $b_{i+1} = {a \bmod {b_i}}$ for $i \geq 0$. Eventually we have $b_{n+1} = 0$; put $P(a,b) = n$.

Example: $P(35, 22) = 7$, since $b_1 = 22$, $b_2 = 13$, $b_3 = 9$, $b_4 = 8$, $b_5 = 3$, $b_6 = 2$, $b_7 = 1$, $b_8 = 0$.

Prove or disprove: $P(a,b) = O((\log a)^2)$.

\end{conjecture}

Keywords: Pierce expansions

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