A scientific lecture at the Research Institute for Mathematics in Oberwolfach, Germany, is no uncommon event. Nevertheless, the lecture by the American mathematician Dan Goldston, of the State University of San Jose, in spring 2003 was of a completely different order. Its contents took the mathematical community by storm. Together with his Turkish colleague Cem Yildirim, it looked like Goldston had advanced attempts to prove the so-called twin primes conjecture by a significant step. What is it about these quirky siblings that so excites mathematicians?

Within the group of integers, prime numbers are in a way thought of as atoms, since all integers can be expressed as a product of prime numbers (for example, 12 = 2 × 2 × 3), just as molecules are made up of separate atoms. The theory of prime numbers continues to be shrouded in mystery and still holds many secrets. Here is just one example: In 1742 Christian Goldbach and Leonhard Euler formulated the unproven Goldbach conjecture, which says that every even integer greater than 2 can be expressed as the sum of two primes (for example, 20 = 3 + 17).

While the chemical periodic table comprises only 10 dozen elements out of which all materials consist, the two ancient Greek mathematicians Euclid and Eratosthenes already knew that there was an infinite number of primes. The most important question is, how are the primes distributed within the system of integers? Taking the first 100 numbers we count 25 primes; between 1,001 and 1,100 there are only 16; and between the numbers 100,001 and 100,100 there are a mere six. Prime numbers become increasingly sparse. In other words, the average distance between two consecutive primes becomes increasingly large (hence the phrase “they are few and far between”).

Around the turn of the 19th century, the Frenchman Adrien-Marie Legendre and the German Carl Friedrich Gauss studied the distribution of primes. Based on their investigations they conjectured that the space between a prime P and the next bigger prime would, on average, be as big as the natural logarithm of P.

The value obtained, however, holds true only as an average number. Sometimes the gaps are much larger, sometimes much smaller. There are even arbitrarily long intervals in which no primes occur whatsoever. The smallest gap, on the other hand, is two, since there is at least one even number between any two primes. Primes that are separated from each other by a gap of only two—for instance, 11 and 13, or 197 and 199—are called twin primes. There are also prime cousins, which are primes separated from each other by four nonprime numbers. Primes that are separated from each other by six nonprime numbers are called, you guessed it, sexy primes.

Much less is known about twin primes than about regular primes. What is certain is that they are fairly rare. Among the first million integers there are only 8,169 twin prime pairs. The largest twin primes so far discovered have over 50,000 digits. But much is unknown. Nobody knows, for instance, whether an infinite number of twin prime pairs exist, or whether after one particular twin prime pair there are no larger ones. Mathematicians believe that the first case holds true, and this is what Goldston and Yildirim set out to prove.

What they claimed was that an infinite number of gaps exist between consecutive primes that are much, much smaller than the logarithm of P, even if P tends toward infinity. The two mathematicians were not given much time to rejoice over their findings. No sooner was it announced than they were awoken to reality. Two colleagues had decided to retrace their proof step by step. In the course of painstaking work, they noticed that Goldston and Yildirim had neglected an error term, even though the term was too large. This was unacceptable and invalidated the entire proof.

Two years later, with the help of Janos Pintz from

Hungary, Goldston and Yildirim revised their work. They managed to plug the hole, and their proof is now believed to be correct. Even though their work does not prove that there are an infinite number of twin prime pairs, it is certainly a step in the right direction.

Working on the theory of twin primes may be more than just an intellectual exercise, as Thomas Nicely from Virginia discovered in the 1990s. Hunting for large twin prime pairs, he was running through all integers up to 4 quadrillion. The algorithm required the computation of the banal expression x times (1/x). But to his shock, when inserting certain numbers into this formula, he received not the value 1 but an incorrect result. On October 30, 1994, Nicely sent an e-mail to colleagues to inform them that his computer consistently produced erroneous results when calculating the above equation with numbers ranging between 824,633,702,418 and 824,633,702,449. Through his research on twin primes, Nicely had hit on the notorious Pentium bug. The error in the processor cost Intel, the manufacturer, $500 million in compensations—a prime example (no pun intended) that mathematicians can never tell where their research, and errors, may lead them.