Annales de l’institut Fourier (AIF)

Kiritchenko, Valentina

Chern classes of reductive groups and an adjunction formula

Mendès France, Michel

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L’INSTITUT FOURIER
Michel MENDÈS FRANCE
Stalking staggeringly large numbers
Tome 56, no 7 (2006), p. 1983-1985.
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Ann. Inst. Fourier, Grenoble
56, 7 (2006) 1983-1985
STALKING STAGGERINGLY LARGE NUMBERS
by Michel MENDÈS FRANCE
This special issue represents the proceedings of a very successful meet-
ing organised by Jean-Louis Verger-Gaugry in 2005 which took place in
Grenoble. The unifying theme was Numeration Systems and this in turn
touches upon a huge variety of topics such as for example Dynamical Sys-
tems, Automata Theory, Fractals and Tilings, Number Theory, Language
Theory and even Physics !
When prefacing a book, I understand it is traditional to survey the dif-
ferent articles it contains. I shall depart from the usual custom for at least
two reasons. First of all, the reader can easily glance through the table of
contents together with all the abstracts both in French and English. Sec-
ondly, I would very much like to take the opportunity of saying a few words
about Jacques Harthong (1948 – 2005) who died recently at the early age
of 57. Very discrete and modest, he never really took pains to advertise
his own work though apart from many papers on mathematics, physics
and even philosophy, he wrote a book on Quantum Mechanics [4] and a
book on Probability Theory [6]. Those who knew his work admired and
praised him greatly for his originality and the depth of his insight. He very
definitely was an exceptional mathematician.
If I mention him here in this Preface, it is because some 20 years ago he
wrote a beautiful paper “Le Continu et l’Ordinateur” [5] which concerns
Numeration Systems and more specifically huge numbers and their repre-
sentation, a topic which more than 2000 years ago Archimedes got involved
in [2].
We all naively think that given any large integer x and its decimal rep-
resentation, we can easily write down those integers less than x. Jacques
1984 Michel MENDÈS FRANCE
Harthong remarks that this is not so. Following D. Knuth, define recursively
a ↑ b = ab
a ↑↑ b = a ↑ (a ↑ b)
a ↑n b = a ↑n−1 (a ↑n−1 b).
For example 2 ↑3 2 = 222
22
(4 exponents). With this notation there is
no problem in writing down staggeringly large integers: x = 10 ↑1000 10.
Now consider the interval [x, 2x] for example. Both extremities are simply
expressed by a string of digits with a huge tail of 0’s. Similarly, there is no
problem in representing x+1, x+2, . . . , 32x, . . . , 2x− 2, 2x− 1. But we will
never be able to fill in the dots convincingly.
The problem is to represent a “generic” number y between x and 2x say.
Its digits form a random sequence à la Kolmogorov. The shortest algorithm
which generates the sequence has precisely as many steps l(y) as the length
L(y) of the sequence. Harthong defines the entropy of an integer z to be
the ratio
l(z)/L(z) 6 1.
He argues that for most integers in [x, 2x] the entropy is maximal whereas
it is practically 0 for expressible integers. Since l(z + 1) ≈ l(z), neighbors
of high entropy numbers have high entropy and low entropy numbers are
surrounded by low entropy numbers so that their set looks like a Cantor
set. Huge expressible integers are therefore extremely sparse. This state-
ment holds whatever the choice of the numeration system. Needless to say,
Harthong’s analysis is much more precise and pertinent.
As sparse as they may be, big numbers occur here and there in the scien-
tific literature. Here is a short pot-pourri. We already mentioned Archimedes.
He estimated the number of grains of sand in the Universe to be something
like 1060, ([2], p. 373). The number of different configurations for a perfect
gas at normal pressure and temperature in an ordinary sized bottle is of
order 1010
23
, ([3], p. 8). The largest known prime number 225964951− 1 was
discovered by Martin Novack in 2005. The Japanese seem to have a word
to express 1088, namely “muryontaisuu” which, I am told by Tadashi Tok-
ieda, means a large number of nothing, ([1], p. 16). I am then reminded
of some medieval philosopher or scientist who claimed that geometrical
points are so incredibly tiny that if you were to bring all the points of the
Universe together, they would easily hold in a thimble. Mathematics meets
surrealism. . .
ANNALES DE L’INSTITUT FOURIER
STALKING STAGGERINGLY LARGE NUMBERS 1985
BIBLIOGRAPHY
[1] J. H. Conway & R. K. Guy, The book of Numbers, Copernicus, Springer-Verlag
ed., 1996.
[2] E. J. Dĳksterhuis, Archimedes, Princeton University Press, 1987.
[3] D. L. Goodstein, States of Matter, Dover, 1975.
[4] J. Harthong, “Études sur la Mécanique Quantique”, in Astérisque, vol. 111, Société
Mathématique de France, 1984.
[5] ——— , “Le continu et l’ordinateur”, L’ouvert 46 (1987), p. 13-27.
[6] ——— , Probabilités & Statistiques, de l’intuition aux applications, Diderot, Arts et
Sciences éd., Bibliothèque des Sciences, Paris, 1996.
Michel MENDÈS FRANCE
Université Bordeaux 1
A2X
33405 Talence Cedex (France)
michel.mendes-france@math.u-bordeaux1.fr
TOME 56 (2006), FASCICULE 7


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Chern classes of reductive groups and an adjunction formula

Stalking staggeringly large numbers

Euler system for Galois deformations

A remark on Whittaker functions on SL $(n,{\mathbb {R}})$

Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry

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