(Conway).
There exists a particular, explicitly constructible function
such that is periodic
for a finite modulus d and takes rational values, for which there is
no Turing machine that, when given n, always decides in a finite number
of steps whether or not some iterate with
is a power of 2.
such that 
is periodic 
for a finite modulus d and takes rational values, for which there is
no Turing machine that, when given n, always decides in a finite number
of steps whether or not some iterate 
with
is a power of 2.