Write
, then
.
Induction on
(one could also use some straightforward way, but I think this way's nicer, and it kills topic 139, too):
We show that it is constant beginning from the
-th term (or earlier):
is clear.
For any
, we write
with
odd.
The sequence
clearly gets constantly
for all
. So we are left to prove the same
.
By induction, the sequence gets constant
. Thus there is
such that for all
we have
.
This gives
by the theorem of Euler-Fermat, meaning nothing else than constantness of the sequence for all
, our result.
