How to check a processor
by Pawel Strzelecki
In 1993 Intel introduced its Pentium processor;
today many personal computers all over the world have it. In the
same year Thomas Nicely, professor of mathematics at the Lynchburg
College (Virginia, USA) decided to improve the available numerical
estimations of the socalled Brun's constant.
What is Brun's constant? A definition first: two prime numbers
are said to be twin, if they differ by two (like 3 and
5, 17 and 19 or p_{1 }= 824,633,702,441
and p_{2 }= 824,633,702,443). The
problem of whether the number of such pairs is finite or not is
still open. The largest twin numbers we know are
.
Shortly after World War I a Norwegian mathematician Viggo Brun
proved that the sum of inverses of all twin prime numbers
is finite. Thanks to numerous fans of mathematics we know today
that B is approximately 1.90216057778 (and this is perfectly
useless knowledge).
However, the most interesting point is still to come. When Nicely
analysed the first results of his computations in June 1994, he
discovered to his own surprise that for some n the resulting
values of (the number
of prime numbers not greater than n) differ from those
that appear in literature. After a long and thorough search for
bugs in the software, after discovering an error in the Borland
C++ compiler etc. it finally turned out that the reason for the
discordance is quite simple: Pentium cannot divide  more
precisely, the values of 1/p_{1} and
1/p_{2} for the twelvedigit twins
quoted above are returned with only nine exact decimal digits
(instead of 19 as promised by the company).
At first Intel showed no reaction to Nicely's news. But
then the news were spread in the Internet, which pushed the company
into an exchange of processors at customer's request. Notwithstanding,
Nicely, who has eventually found 135,780,321,665 pairs of twin
primes in the interval [1, 10^{14}],
employs at least two different computers for every computation
ever since.
Nicely's story is just a coincidence. Nevertheless, quite often
large programs designed for searching for large primes (or finding
distant digits in the decimal expansion of )
are used by hardware producers to test their products. For instance,
the then largest prime number announced in September 1996, i.e.,
, was discovered during tests performed
on the new CRAY.
