Twin primes are pairs of primes that differ by 2. For example, 3 and 5 are twin primes, as are 17 and 19. Importantly, so are 824633702441 and 824633702443. More on that in a minute.

No one knows whether there is a largest pair of twin primes. The twin prime conjecture says that there are infinitely many pairs of twin primes, but the conjecture has not been proven.

Now suppose we take the reciprocals of the twin primes and add them up.

If there were only finitely many twin primes, the sum would have finitely many terms and hence a finite sum. But the sum might converge even though it has infinitely many terms. On the other hand, if we could show that the sum diverges, we’d have a proof of the twin prime conjecture. Viggo Brun showed that the sum does converge. Its sum, known as Brun’s constant, is a little more than 1.9.

In 1994, Thomas Nicely was studying Brun’s constant when he found that his computer incorrectly computed 1/824633702441 beyond the eighth significant figure. Nicely had discovered the infamous Pentium division bug.

Intel responded by saying the division errors were inconsequential. Intel was absolutely correct, but the public couldn’t understand that. They only knew that the chips were “wrong.”

The error was estimated to occur once in every 9 billion divisions. (I doubt any large program has ever been written that is as bug-free as the buggy Pentium chips.) And when an error did occur, the result was not entirely wrong, only less accurate than usual. The public only understood that sometimes the answers were “wrong.” Most people do not understand that floating point arithmetic is nearly always “wrong” in the sense of being less than perfectly accurate.

At first Intel said it would only replace the chips for people who could show they were effected by the bug, i.e. almost nobody. Eventually Intel gave in to pressure and replaced the chips. The episode cost Intel half a billion dollars.

More number theory posts

