A discrete iteration related to Pierce expansions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For integers $a>b>0$, set $b_1=b$ and, whenever $b_i>0$, let $b_{i+1}$ be the least nonnegative remainder of $a$ on division by $b_i$. The numerator $a$ remains fixed throughout. Since $0\leq b_{i+1}<b_i$, the sequence reaches zero. Define $P(a,b)=n$ when $b_1,\ldots,b_n$ are positive and $b_{n+1}=0$. Here $\log$ is the natural logarithm.

Statement recorded in the supplied source.

Is there an absolute constant $C>0$ such that $P(a,b)\leq C(\log a)^2$ for every pair of integers $a>b>0$? [S2]

Short English statement

Does repeatedly replacing the current denominator by the remainder of one fixed numerator always reach zero within a constant times the square of the numerator logarithm?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3348 (OPG-819). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, A discrete iteration related to Pierce expansions, node 819, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/a_discrete_iteration_related_to_pierce_expansions.