5, 11, 17, 23, 29 is an arithmetic progression of prime numbers. It is conjectured that there are such progressions with arbitrarily many terms (there cannot be infinite sequences because a, a + d, a + 2d,… sooner or later contains a + ad). The longest such known to the authors has first term 4943 and common difference 60 060 with 13 prime terms, but longer ones will have been found by now. In general, additive problems about the multiplicatively defined prime numbers are inevitably difficult (e.g. the Goldbach conjecture and twin primes) and may even be candidates for undecidability. We have been trying to prove a much more general conjecture of Erdos, spurred on by his offer of $3000 to the first proof or disproof of it.

It was quite clear that the company were much more interested in what the Baron had to say about the marriage customs of the people of Octonaria than in his remarks about their system of numbering. So I bided my time until he left the hospitable parlour of The Dolphin and, attaching myself to him, with little difficulty persuaded him to accompany me to my lodgings. Here, comfortably ensconced in my own armchair, he proceeded to fill the great octagonal pipe which had excited such curiosity at The Dolphin, while I set about collecting the ingredients for a bowl of punch.

The sequence 1, 3, 7, 13, 21, … arises in many ways, for example from the integers placed in a spiral on a square grid, the terms arising where the spiral intersects a diagonal on the grid, and as the central axis of a triangular array of numbers:

A bag contains m > 0 black balls and n > 0 white balls. A sequence of balls from the bag is discarded in the following manner:

• (i) A ball is chosen at random and discarded.

• (ii) Another ball is chosen at random from the remainder. If its colour is different from the last it is replaced in the bag and the process repeated from the beginning (i.e. (i)). If the second ball is the same colour as the first it is discarded and we proceed from (ii). Thus the balls are sampled and discarded until a change in colour occurs, at which point the last ball is replaced and the process starts afresh.