's and 
's representing the branches
taken by a number during iteration of the
function he uses,
Recall that Terras' parity vector
is a sequence of 's and 's representing the branches
taken by a number during iteration of the
function he uses,
At each iteration,
the parity vector 
of a number 
contains a if the number at this stage is odd
(so that the 
rule is used), or it contains a
when the number is even (and the 
rule applies):
where 
is the 
-th iteration of the function .
We take 
, and 
.
If a number is of the form 
,
with 
an odd integer and 
,
then 
for 
, and 
.
Moreover, 
.
because the original number is odd.
Then the rule is applied, resulting in
which has a form similar to the original ,
but with one less factor 
and one more factor 
.
As long as the exponent of is larger than zero,
we will continue having an odd number ()
and be able to apply again the rule.
After 
iterations, the exponent of will be consumed,
and the number replaced by 
, which is even because is odd;
hence .
Note that a number of the form , with odd, has a binary representation
that ends in bits with a value of 
, followed to the left by a bit with a value of 
.
(Because the binary representation of 
will have bits at the end,
and then the ending bit of , which is because is odd;
adding to toggles these bits.)
This means that the parity vector of coincides (if laid out right-to-left)
with the last 
bits of the binary representation of the number.
Thus we know, for example, that the number 
(
in binary) will take the branch
three times before using the rule, because 
.