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Pierce expansions
A discrete iteration related to Pierce expansions ★★
Author(s): Shallit
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: .
Keywords: Pierce expansions
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