I wonder if any other math-funners have been the recipients of a curious example of mathematical spam, recently deposited in my in-box with the customary ominous thud. Among other things, its covering gloss observes that the (putative) author has spent some 30 years perfecting his solution to the "problem of finding exponents of prime numbers", and invites the reader's opinion of the attached 47-page treatise on the topic. Painstaking inspection of this document eventually reveals that the "exponent" under discussion is (minus) the p-adic valuation m! [the maximum power of p dividing factorial m]. I have on reflection decided that it might be both kinder and wiser to refrain from suggesting that he consult Graham, Knuth & Patashnik's "Concrete Mathematics", which I believe quotes a theorem of Legendre from 1830 to the effect that this exponent equals (m - s_p(m))/(p - 1), where s_p(m) denotes the sum of the digits of m in base p. But I can't help feeling that it is a great shame he could not have devoted just a few weeks of his 30 years to reading this wonderful book, from which he might just have actually learnt some beautiful and useful mathematics; instead of which, he seems frozen in a state of perpetual numerical infantilism. "A little knowledge is a dangerous thing ..." ----------------------------------------------------------------------- Oddly, I had also been contemplating exactly the same problem for the previous few days; or at any rate an extension of it. Several authors have investigated divisibility properties of Stirling numbers of the second kind S(n, m); but what about the first kind, s(m, n)? Since s(m, 0) = m!, it's reasonable to enquire whether Legendre's classic result might generalise. The motivation for looking at this stems from divisibility of Bell numbers B(n). From the combinatorial definitions, B(n) = \sum_m S(n, m), for n >= 0. And extending the Stirling number recurrence S(n, m) = S(n-1, m-1) + m*S(n-1, m) backwards reveals the little-known law S(-n, -m)*s(m, n) = (-1)^(n+m), permitting both kinds to be defined for all n,m [but don't tell Maple, which insouciantly returns zero for all negative first arguments]. Feeding back to the first equation then suggests defining B(n) = \sum_m |s(m, -n)|, for n < 0. The sum is plainly infinite in the usual metric, but converges p-adically for all p, with m terms giving in practice nearly m/(p-1) base-p digits. Now B(n) is already known to be periodic mod p^t for n >= 0; with the extension, the periodicity appears to extend to all n. Using the fact that S(n, n-k) (extended) is polynomial in n of degree 2*k, B(n) can be extended to a continuous function of a continuous variable; p-adically this function is almost-periodic. Fred Lunnon [03/08/07]