Mathematics at all levels depends on infinite processes. This is a “fact of life”. But, like those other “facts of life”, this one has traditionally been hushed up—especially when teaching elementary mathematics to beginners. Of course, in our liberated age, adult mathematicians are no longer embarrassed by infinite processes, and even discuss them openly. But it is not easy to see the connection between the way they discuss infinite processes (as in, say, an undergraduate analysis course) and those parts of elementary mathematics which depend on infinite processes, and which should provide the soil in which the beginner’s seedling intuitions about infinite processes can take root. So even today most children, adolescents, and their teachers must either make do with the traditional smutty versions, or work things out for themselves.

In a recent issue of the Gazette, S. Parameśwaran introduced S·P numbers. A positive integer n is an S·P number if it equals the sum s of its digits multiplied by the product p of its digits, i.e. n = sp. Subsequently, H. J. Godwin showed that in any number base the number of S·P numbers is finite, and A. F. Beardon and K. Robin McLean enumerated all S·P numbers in base 10 (namely, 1, 135, and 144). We generalise Beardon’s technique from base 10 to an arbitrary base to determine all S·P numbers in bases 2 through 12. Contrary to the claims of Godwin, we find 4-digit and 5-digit S·P numbers in base 11.

(In reference to A. C. Dixon’s note on lm + 2m + 3m +...+nm, on pp. 283-4 of this volume, the following further notes may be of interest. They are taken, with some alterations, from a paper communicated to the London Mathematical Society in February, 1910.)

The Report was prepared at the request of the Sub-committee on Instruction of the British National Committee for Mathematics. It formed the basis of the presentation of the United Kingdom point of view in a discussion on the topic at the International Congress of Mathematicians held in Moscow in August, 1966.