Those Fascinating Numbers 407

243 112 608(243 112 609

− 1)

• the

46th

even perfect number, a 25 956 377 digit number.

3 · 2402 653 211 − 3

• the number of steps required for the Goodstein sequence starting at 4 to reach

0: this is a 121 210 695 digit number; in 1944, the English logician R.L. Good-

stein introduced an algorithm to generate sequences of positive integers which,

contrary to what we may think, converge to 0; to describe the process put forth

by Goodstein, we introduce the notion of “complete representation” of a posi-

tive integer in base b: first write this number as a sum of multiples of powers

of b, and then do the same with the exponents found in this sum, then the ex-

ponents of these exponents, and so on, until the representation becomes stable;

for example, the complete representation in base 2 of 266 (= 28 + 23 + 21) is

222+1

+22+1 +21;

thus, the Goodstein process for the number 3 is the following:

3 =

21

+ 1 ·

20

−→

31

+ 1 ·

30

− 1 = 3 = 1 ·

31

−→ 1 ·

41

− 1 = 3 = 3 ·

40

−→ 3 ·

50

− 1 = 2 = 2 ·

50

−→ 2 ·

60

− 1 = 1 = 1 ·

60

−→ 1 ·

70

− 1 = 0;

Goodstein [94] proved in 1944 that every Goodstein sequence converges to 0;

this fact is somewhat counter intuitive. Indeed, although the sequence obtained

by beginning with 3 converges rapidly to 0 (in only five steps), a very different

situation occurs by starting at 4, since the number k of steps required by the

Goodstein process to bring 4 down to 0 is k = 3 ·

2402 653 211

− 3 (L. Kirby and

J. Paris [119])215.

10101034

• the Skewes number; this number occupies an important place in the history

of the function π(x): indeed, in 1933, assuming the Riemann Hypothesis,

Skewes proved that the smallest number x0 for which π(x0) Li(x0) satis-

fies x0

10101034

; this result was at that time very significant, since 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 1023,

but which is not always true; indeed, Littlewood proved in 1923 that the differ-

ence π(x)−Li(x) changes signs infinitely often; in fact, he proved more, namely

that there exists an increasing sequence of real numbers x0, x1, x2, . . . tending

215Kirby

and Paris established that the very slow convergence of the Goodstein sequences to 0 is

related to the Goodstein Theorem which cannot be proved within the setup of elementary arithmetic.