(Conway). For every partial recursive function f defined on a subset D of the natural numbers there exists a function such that(1) is periodic for some d and takes rational values.
(2) There is some iterate such that for some j if and only if m is in D.
(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.