![](/files/happy5.png)
A discrete iteration related to Pierce expansions
Conjecture Let
be integers. Set
and
for
. Eventually we have
; put
.
![$ a > b > 0 $](/files/tex/db6c431d4c315f8358aa8c7f6636f0ac24e2f729.png)
![$ b_1 = b $](/files/tex/4b560d3e5d5afa071b053545576a417c57383a7c.png)
![$ b_{i+1} = {a \bmod {b_i}} $](/files/tex/4583b01aed3106f42d256c750b0b25d45cbdf594.png)
![$ i \geq 0 $](/files/tex/3cce6a1da7b5ac02ce84bc38a57ce00ec233239d.png)
![$ b_{n+1} = 0 $](/files/tex/3a01335006332daff6d25c3eeef378d060aa1a3e.png)
![$ P(a,b) = n $](/files/tex/857ed2344d57a36f86a8013227fe025c5e2793ac.png)
Example: , since
,
,
,
,
,
,
,
.
Prove or disprove: .
The best upper bound is currently . For more information, see [ES].
Bibliography
[ES] P. Erd\"os and J. Shallit, ``New bounds on the length of finite Pierce and Engel series'', S\'eminaire de Th\'eorie des Nombres de Bordeaux 3 (1991), 43--53.
* indicates original appearance(s) of problem.
A different upper bound
This paper shows an upper bound of
.
Edit: But looking at the title page of the paper, I see you already knew that ;)