### Les nombres remarquables

As I was preparing the lecture I just gave for the SMO Day, I came back to a book I had loved when I was a kid but which I had not spent much time reading since, François le Lionnais’s *Les nombres remarquables*, written with J. Brette (this is a title which is surprisingly hard to translate; “Remarkable numbers” doesn’t seem right, because it misses the subtle highlighting *Les*).

In any case, this is a kind of dictionary of interesting numbers, which was published in 1983. I know there exist a few others, but this is the only one I’ve looked at, so I can’t compare it with them. However, this is definitely a very nice book. Partly this may be due to the fact that le Lionnais (should one say, the author, or the collector?) was not really a professional mathematician (despite what Wikipedia seems to claim in the English article), but a type of renaissance man interested both in sciences in general and a humanist; he was the co-founder (with R. Queneau, the man behind *Zazie dans le métro*) of the famous *Oulipo*. However, there is no cute amateurishness in his selection of numbers; of course, he doesn’t disdain curiosities like

843 973 902. The largest number obtained, in the decimal system, as a product of two numbers constructed with all digits from 1 to 9. 843 973 902=9642 x 87531.

but what he (or they) think are the “most interesting numbres” are highlighted with one to three stars, and the next item shows that they know that mathematics is not all cuteness:

898 128 000. *** 2^{7}3^{6}5^{3}7 11. Order of the McLaughlin group, the tenth sporadic group.

There are even a few complex numbers, and some “non-determined finite numbers”. Among the first, I note the following property of

which I didn’t know (or had forgotten): the triples

are the only triples of (distinct) complex numbers with modulus one such that 0 does not belong to the convex hull of

for any integer *n*.

Among the non-determined finite numbers, there is the one defined as follows, which I suspect is probably determined now:

where *k* is the digit following the first sequence of seven consecutive sevens in the decimal expansion of *π*. (I think sufficiently many digits have been computed now to determine *N*).

Paul — 28.03.2009 @ 18:07

N == -1000

You can search it here: http://www.angio.net/pi/bigpi.cgi

I am not sure about the value of E^(2 I Pi/7) given, it looks awfully close to 1…

Kowalski — 28.03.2009 @ 18:17

Cool… I didn’t find a search-Pi-page with my own 30 s web search…

And you’re correct of course for exp(2i pi/7)! The real value is

0.62349 + 0.78183*I

and I really wonder where the value in the book comes from. It doesn’t seem to be a problem of computing in degrees or radians. (I don’t even known if the stated property is valid… there is no reference…)

In fact, the very short section on complex numbers has serious problems: two other items are Gamma(1+i), where the sign of the imaginary part is wrong, and exp(i), where the decimal sign is missing in the imaginary part!

Anonymous — 29.03.2009 @ 0:17

Dear K,

I think you will find that Cos(Pi^2/630) = 0.99987729…

To put it another way, this a version of a classic “calculator error”: computing Cos of 2 Pi/7 _degrees_ rather than _radians_.

Kowalski — 29.03.2009 @ 9:00

I had suspected the radian/degree problem (the teaching in France used degrees to measure angles in a lot of elementary geometry at that time — I don’t know about now), but I would never have guessed that whoever did the computation would have replaced 2*Pi par Pi^2!

(And the imaginary part also corresponds to sin(Pi^2/7/90).)

Anonymous — 29.03.2009 @ 20:05

Dear K, what I meant was 2Pi/7 degrees is equal to (2Pi/7)(2Pi/360) = Pi^2/630 radians. So if you have one of those calculators that assumes (unless you change the settings) that any input is in degrees, and you type 2 x Pi % 7 Cos, this is what comes out.

Kowalski — 29.03.2009 @ 20:18

Thanks (again…) I should have used a calculator instead of messing up the radian-to-degree conversion by putting a single Pi myself…

So the mystery is explained.