How do These Series End?
In the earlier chapters we have already encountered many sequences of numbers which may or may not go on forever. Particularly famous examples are the perfect numbers and the amicable numbers, and others, somewhat less well known, are the emirps and the prime twins. Obviously these sequences are in actuality either finite or they are not; the uncertainty simply reflects our present lack of knowledge concerning them. In some rather embarrassing cases, such as the odd perfect numbers, odd weirds, and crowds, we do not even known whether the sets possess any members at all, although they could quite easily contain an infinite number. Such is the degree of our ignorance. In this chapter we focus on these kinds of difficulties by setting up even more simply defined sequences of integers and asking once again whether these series go on forever, or whether at some point they stop. In some, as is the case with the perfect and amicable numbers, the answers remain unknown, while in others some insight has been achieved.