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 so-called 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 p1 = 824,633,702,441 and p2 = 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/p1 and 1/p2 for the twelve-digit 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, 1014], 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.