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 ..."
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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]
