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.
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