A first observation is that there are other cycles if negative integers are
allowed in the domain of the function.
There is a cycle of period 1 starting from n = -1,
and there are cycles of length 3 and 11 starting from n = -5 and n = -17,
Böhm and Sontacchi  conjecture that these cycles together
with the cycles starting with n = 0 and n = 1
make up the entire set of cycles occurring under
iteration of applied to the integers .
Several authors have proposed the following conjecture (, , , ).