Today's lecture at the Des Mathématiques
seminar (where researchers try to explain some interesting or fun part
of mathematics at a level understandable by first-year students) was
rather amusing, so I'm inclined to say something about it. It
revolved about the study of the decimals (or, more generally, base
`b` digits) of 1/`p` where `p` is a prime
number.

Here's one basic observation: for `p` not equal to 2 or 5,
the first `p`−1 decimals form a period block (which is
then repeated indefinitely). Computing the `p`-th decimal is
therefore very easy: so long as `p` is at least 11, it is
always 0 (because the first decimal is then 0); a little bit less
obvious is the fact that the (`p`−1)-th decimal (the
last in the “obvious period”) is always either 1, 3, 7 or
9 (always for `p` not equal to 2 or 5, of course); as a
matter of fact, it is 1, 3, 7 or 9 according as the last digit of
`p` is 9, 3, 7 or 1 (certainly a prime number other than 2 or
5 cannot end by any other digit, in base 10): so we can immediately
tell that the 18th and 19th decimals of 1/19 are 1 and 0. Computing
the (`p`+1)-th or (`p`−2)-th decimals, say, is
not much more complicated (for `p` at least equal to 101, the
(`p`+1)-th decimal is always 0, and as to the
(`p`−2)-th, it can be computed by dividing
10^{p−1}−1 by `p` in increasing powers
rather than decreasing).

More subtle, and more interesting also, is the following
observation: if `k` is (`p`−1)/2 then the
(`k`+1)-th decimal of 1/`p` (so long as `p`
is at least 11) is always either 0 or 9. Furthermore (and here is
where I leave this as an exercice to those who know the Law of
Quadratic Reciprocity—I guess this is the nicest example that I
know of where Quadratic Reciprocity comes in a very down-to-earth
problem), whether it is 0 or 9 depends only on the remainder of
`p` modulo (= when dividing by) 40, or, more concretely, of
the last three digits of `p`. And the previous decimal, the
`k`-th one, can never be 4 or 5, and it is even when the
(`k`+1)-th is 9 and odd when the (`k`+1)-th is 0.
Can you prove all of this? (The hardest part is the one about
depending only on `p` mod 40, which involves Quadratic
Reciprocity; the rest is all done by elementary congruences.) Again,
we could go on to speak about the (`k`+2)-th and
(`k`−1)-th decimals.

This is an interesting way to begin all sorts of more subtle
observations on arithmetic. For example, I could mention primality
tests: as I said, if `p` is prime (other than 2 or 5) then
the decimal expansion of 1/`p` has period
`p`−1 (I'm not claiming this is the shortest period,
but it is *a* period), so for example 21 is not prime because
its expansion is not 20-periodic (its minimal period is 6). The
converse is not true: for example, 1/1729 is periodic with period
dividing 1728 (actually 18), and in fact this is true not only in base
10 but in *any* base (except 7, 13 and 19, for obvious reasons)
but 1729 is not prime (it is the product of 7, 13 and 19); this is due
to 1729 being a so-called Carmichael number (or
“pseudoprime to any base”). But even the remark about the
digits halfway through the period can detect non-primality: for
example, the decimal expansion of 1/259 has period dividing 258
(namely, 6), so we could think that 259 is prime, but the fact that
the 129th digit is 8 (and not 0 or 9) betrays the non-primality of 259
(which is 7×37). This is actually closely related to the Miller-Rabin
(probabilistic) primality test.

Another natural question would be: is it true that for all
rationals `α` and `β` (and any base
`b`>1) there exists a positive integer `N` such
that, for primes `p` for which
`α`·`p`+`β` is an integer (if
any!) that digit (in base `b`) of 1/`p` depends only
on the congruence class of `p` mod `N`, at least for
large enough `p` (where large enough

may perhaps
depend on `α` and `β`, and certainly of
`b`). This seems plausible, but I haven't given it any
thought (except very briefly when `α` only has a power
of two in the denominator, in which case I believe it follows from
quadratic reciprocity, whereas other cases might involve higher
reciprocity laws).

Finally, the question of the smallest period of the decimal (or,
more generally, `b`-adic) expansion of 1/`p` raises
the question of whether there exist infinitely many primes
`p` such that 10 (resp. `b`) is primitive (which
amounts to the period being `p`−1). I don't remember
what is known about that question: I vaguely recollect Heath-Brown
having proved something remarkable in that direction.