One of the difficulties encountered by students when they are first introduced to axiomatic algebra is that the axioms, such as commutativity, associativity, and distributivity, seem self-evident in the algebraic situations with which they are familiar. Indeed, the axioms are so useful, of course, because of their wide applicability. However, it can be instructive to study situations in which these axioms break down, particularly if the situations are not too contrived.

If p is a prime and ps is the greatest power of p that divides N!, then

Dr. C. D. Langford recently sent me some numerical work he had done on automorphic numbers. He found automorphic numbers in the scales of 6, 10, 12, 14, and 15 and he discovered what proved to be a general method of finding an automorphic number of r + 1 digits from one of r digits. Dr. Langford’s notes stimulated me to look for explanations of his results.

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.

The term measuring number is acquiring currency as a popular synonym for non-negative real number. This is good. Yet the belief persists that the real numbers, by whatever name, must remain fundamentally inaccessible to school children, obscured behind a veil of Dedekind cuts or Cauchy sequences. (Thus Griffiths and Howson, in what is generally a humane book [1], devote several pages to an ingenious, but ultimately condescending, attempt to foreshadow the mysteries of real numbers in terms of points of a line.)

The regular decagon will be dissected into other regular polygons and into Greek and Latin crosses by the methods outlined in my previous article (Math. Gaz. 45, 1961, p. 94).

The first step is to find tessellations. In one of those that are used in this note, shown in 1 and 2, the decagon is cut into three pieces, and in the other only two. A further tessellation element, not used here but worth noting, is shown in 5.