> I implemented some cycle finding algorithm, a tortoise and hare variant maybe but I can not tell from the top of my head, on the reminder sequence giving up after a user-specified number of steps.
What's the cycle finding for? Isn't it enough to record the remainder sequence with indices? As soon as any remainder reoccurs, that's the cycle.
You could of course just maintain the set of remainders seen so far and then you would detect the cycle by reaching a remainder that is already in your set of remainders seen so far but that requires space proportional to the number of digits expanded and if you are expanding a lot of digits and have a large denominator, this will add substantially to the amount of memory consumed. The digits themselves require of course a linear amount of space, too, but only about one byte per digit compared to the potentially big remainder.
What's the cycle finding for? Isn't it enough to record the remainder sequence with indices? As soon as any remainder reoccurs, that's the cycle.