Consider the question of whether a Turing Machine T, on any input for which it does halt, always leaves behind an odd number of symbols on the tape. (Note that we are not saying that T halts on all inputs, or that it halts on any inputs at all, but simply that, if it does halt, there will be an odd number of symbols left on the tape).

Prove by reduction that L_odd, the set of TMs that never halt leaving an even number of symbols on the tape, is not recursive.