Book Review Those Fascinating Numbers Jean-Marie De Konick ...

Book Review
Those Fascinating Numbers
Jean-Marie De Konick.
American Mathematical Society, 2009, 426 pages.
ISBN 978-0-8218-4807-4
2000 Mathematics Subject Classification. 11-00, 11A05, 11A25
This is an unusual book which was originally published in French by the editorial
ELLIPSES under the title: Ces nombres qui nous fascinent 2008 Édition Marketing S.A. It
consists of a list of positive integers and some properties characterizing them, together with
many interesting references which are helpful if you wish to learn more or to investigate
further. In my opinion, this last part is the most interesting feature of the book.
For example, the integer 4 is defined as the Pythagoras number of Z, that is, the
smallest r such that every positive integer is the sum of r squares of integers, while the
integer 9 is defined as the smallest r such that every positive integer is the sum of r cubes
of integers. Both are particular cases of Waring’s important problem: to determine the
smallest g = g(k) such that every positive integer is the sum of g k-th powers. It is
conjectured that g(k) = 2 k + [(3/2) k ] − 2, where [x] stands for the largest integer ≤ x.
Indeed, Pillai proved in 1936 that if 3 k = q2 k + r with 0 < r < 2 k and q + r ≤ 2 k , then
the conjecture holds; for example g(4) = 19.
Some numbers are defined in the book as solutions of numerical equations. For example,
if φ(n) is the number of integers 0 < m < n such that gcd(m, n) = 1 (the Euler
function) and τ(n) is the number of positive divisors of n, including 1 and n, it is an
elementary exercise to determine the solutions of the equation φ(n) = τ(n), namely:
1, 3, 8, 10, 24 and 30. This is used by the author to define 30 as the largest number n such
that φ(n) = τ(n), while 8 is the third of such solutions. Obviously this kind of properties
is not the most impressive in the book.
There are many examples of numbers defined as the largest one satisfying some arithmetical
property; for instance, 33 is the largest integer which cannot be written as the sum
1

of five non zero squares, the other ones being 1, 2, 3, 4, 6, 7, 9, 10, 12, 15 and 18. It has been
more surprising for me to find out that 77 is the largest integer which cannot be written
as the sum of positive integers whose sum of reciprocals equals 1. For example,
78 = 2 + 6 + 8 + 10 + 12 + 40, and 1 1 1 1 1 1
+ + + + + = 1.
2 6 8 10 12 40
Indeed what is deeper is that, in both cases, almost all positive integers satisfy the studied
property.
The last number studied in the book is s = 10101034 , known as the Skewes number. To
explain its interest some notation is needed. Let π(x) be the number of prime numbers
≤ x, and consider the so called logarithmic integral
Li(x) =
x
0
dt
log t .
Many great mathematicians, Gauss and Riemann being two of them, believed that π(x) <
Li(x) for all x ≥ 2, an inequality which can be verified for all x < 10 23 , but which is not
always true. Indeed, Littlewood proved in 1923 that the difference π(x) − Li(x) changes
its sign infinitely often. Skewes proved in 1933, assuming the Riemann hipothesis, that
the smallest number x0 for which π(x0) > Li(x0) satisfies x0 < s, a result which was
considered very significant at that moment. In fact, as far as I know, the exact value of
x0 is unknown.
As expected, other examples in the book are not so exciting. For example, the integer
11 is the smallest prime number p such that 3 p−1 ≡ 1 mod p 2 , and 43 is the fourth prime p
such that 19 p−1 ≡ 1 mod p 2 but the author does not give extra information to understand
why congruences of the form r p−1 ≡ 1 mod p 2 are interesting.
In my opinion, this is a book for browsing rather than for sustained reading, but all in
all it introduces interesting notions and problems in number theory.
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