Sometimes nǃ is very divisible by n. For example, 12ǃ is divisible by 125 while 720ǃ is divisible by 720178. On the other hand, if p is prime then pǃ is divisible by p but not by p2.

In general it is quite difficult to calculate the maximum exponent m(n) such that nm(n) divides nǃ

In this short article I will locate some values of n, for which nǃ is very divisible by n, find a tight upper bound for m(n), and point the reader in the direction of related problems.

A-level courses and examinations are with us for some time to come. Even if Mevel is introduced, for most pupils who spend two years in the I sixth form A-levels will remain the basis of their studies. It seems to me that we need a major debate about A-level mathematics; we need to review it as it is and try to decide what developments are necessary. The discussion I paper ‘Mathematics at A-level’ has already raised a number of points, some of which are touched on here. This article is not a statement of official SMP policy; any opinions expressed in it are my own. The aim is to provoke discussion. I shall consider a number of issues which seem to me to be important, hoping that others will be stimulated to contribute comments and suggest other topics which need attention.

I recently stumbled on an identity (Theorem 2) that related two types of permutations. It looked like one of those serendipitous pieces of mathematics that led to a simple enumerative result; mostly when this happens it leads joyously to an enumerative proof that was ‘blindingly’ obvious. Except in this case, it wasn’t, I don’t think.

It has been known for some time that when the sun is reflected in a circular cylinder, the illuminated part of the cylinder's base is bounded by part of a nephroid (the curve obtained by rolling a circle around another circle with twice the radius).

I have long been convinced that the teaching of mathematics suffers from the premature introduction of abstraction without adequate attention to the concrete facts from which the abstractions are derived. Perhaps I cannot better begin my argument than by taking as text a passage from the end of the first chapter of Todhunter’s Differential Calculus :

It is only after considerable hesitation that I have allowed myself to be persuaded to publish this paper, for it contains nothing whatever that cannot be obtained from any of the standard text-books on the Theory of Numbers. I have, however, been influenced by the fact that to many school-masters such text-books are not easy of access, and that this is likely to be so more particularly in the case of those who were unable to attend the January meeting. Thus, although I feel that the pages of the Mathematical Gazette are hardly the place for the reproduction of selected passages from standard mathematical works, I have in this case allowed my scruples to be overcome, in the hope that the concession may prove useful to those who have neither the leisure nor the opportunity to become acquainted with the subject from the pages of Mathews or Dickson.

For a long time prime numbers have attracted the attention of mathematicians, especially those primes that possess some sort of a symmetry. The mysterious repunits An = 111… 1(n), whose decimal representations contain only units, form an important class of them. For a repunit to be prime the number n of its digits must be also prime. But this condition is far from being sufficient: for instance, A3 = 111 = 3.37 and A5 = 11111 = 41.271. Some of the repunits are nonetheless prime: A2, A19, A23, A317 and A1031, are the only known examples. The question of primeness of the repunits was discussed by M. Gardner and later in [2–4]. It is not clear whether the number of prime repunits is finite or infinite.