once mentioned to Ramanujan that the number of the
taxicab he had ridden that day was 1,729, a number
that he thought was “rather dull.” Ramanujan imme-
diately responded, to the contrary, that 1,729 is not
at all dull, being the first of all the positive integers
that can be written as the sum of two cubes in two
different ways. In an instant, Ramanujan somehow
realized that 1,729 equals both 123+ 13and 103+ 93,
and that no other number smaller than 1,729 has this
dual property.
Over the 5 years of their time together in England,
Hardy and Ramanujan produced a number of funda-
mentally important results in number theory. Their
most notable work was on the theory of
PARTITION
s.
During this time Ramanujan was awarded a bachelor
of science by research degree (the equivalent of a Ph.D.)
from Cambridge, was elected a fellow of the R
OYAL
S
OCIETY
of London—England’s most prestigious scien-
tific body—and also a fellow of Trinity College.
Unfortunately, Ramanujan suffered from extremely
poor health. After developing tuberculosis, Ramanujan
returned to India in 1919 but died 1 year later on April
26, 1920.
Ramanujan left behind him a number of unpublished
notebooks filled with theorems and claims that mathe-
maticians have continued to study. Mathematics profes-
sor George Neville Watson (1886–1965) studied the
notebooks and published 14 papers, all under the title of
Theorems Stated by Ramanujan. He also published an
additional 30 papers, all inspired by Ramanujan’s work.
random numbers A sequence of numbers with the
property that no next number in the sequence can be
predicted from the preceding elements is said to be ran-
dom. Such a sequence does not follow any regular or
repetitive pattern. If the numbers listed come from a
finite pool of possible candidates (say, single-digit num-
bers 0 through 9), then the
LAW OF LARGE NUMBERS
asserts that, in the long run, each number from that
pool should occur equally often.
The traditional method for generating random
numbers is to draw numbered balls from a container.
It is not possible to generate truly random quantities
with a computer—any program is a predetermined set
of instructions—but it is possible to create a list of
numbers that appear to be random. Numbers gener-
ated in this way are called pseudorandom. The follow-
ing are two popular methods for generating pseudo-
random numbers.
Middle-Square Method
Developed in 1946 by J
OHN VON
N
EUMANN
(1903–57)
and his colleagues while working on the Manhattan
Project at Los Alamos Laboratories, the middle-square
method works as follows:
1. Select a four-digit number to be the first number in
the sequence. (This number is called the seed of the
algorithm.)
2. Square this number to produce an eight-digit num-
ber. (Add a leading zero if necessary.)
3. Use the middle four digits of this eight-digit number
as the next number in the sequence. Repeat.
The result that appears is a seemingly random list of
numbers between 0 and 9999. For example, beginning
with the seed 7,254 we obtain the sequence 6205, 5020,
2004, 1601, 6320, 9424, 8117, … Unfortunately, the
middle-square method can produce sequences of inte-
gers that tend toward zero. For example, beginning
with the seed 1049 we obtain the sequence 1049, 1004,
80, 64, 40, 16, 2, 0, 0, 0,…
Linear-Congruence Method
Developed by D. H. Lehmer in 1951, the linear-congru-
ence method uses
MODULAR ARITHMETIC
to generate a
list of pseudorandom numbers. It works as follows:
1. Select three fixed numbers a, b, and m, and an ini-
tial value x0(the seed).
2. Given a number xnin the sequence, the next number
xn+1 in the list is obtained by multiplying xnby a,
adding b, and taking the remainder upon division
by m:
xn+1 = axn+ b(mod m)
For example, taking a= 2, b= 3, and m= 10, with seed
x0, = 1, produces the sequence 1, 5, 3, 9, 1, 5, 3, 9, 1,
…., which, unfortunately cycles and is far from pseu-
dorandom. Although it is not possible to avoid cycling
with this method, one can choose an extraordinarily
large value for mso that the length of the cycle pro-
duced is extremely long, and the repetition of numbers
will not be encountered in a lifetime. (This creates a
436 random numbers