# A discrete iteration related to Pierce expansions

 Importance: Medium ✭✭
 Author(s): Shallit, Jeffrey O.
 Subject: Number Theory
 Keywords: Pierce expansions
 Prize: \$50 USD
 Posted by: shallit on: June 11th, 2008
Conjecture   Let be integers. Set and for . Eventually we have ; put .

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 ;)

### bound

That's because the best currently known bound is not the one in the paper. It is in a technical report by Vlado Keselj.