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.
Odd perfect numbers
Probability that a number is prime
CHI Conversations has posted a talk by Jef Raskin, designer of the first Macintosh, entitled Macintosh: Lessons Learned, Lessons Lost. The talk was recorded in 2004, about a year before Raskin’s death. Much of the talk is devoted to Macintosh history and how inaccurately it has been reported.
Toward the end of the talk, around 55 minutes in, Raskin compares Mac and Windows.
The Mac has gotten harder to use over the years. In fact, Windows has gotten easier. Now I can move back and forth, I can hardly notice the difference. Mac is just epsilon easier to use.
Raskin complains that software bloat has outpaced Moore’s Law, a contention that was verified here.
Jef Raskin. Photo by his son Aza.
Moore’s Law and software bloat
Inside Steve Jobs’ Brain
People have been comparing software components to LEGO blocks for a couple decades. We should be able to assemble applications by snapping together modular components, just like LEGOs. There has been progress, but for the most part we haven’t realized the promise LEGO-style software development.
Integrating two software systems is usually more like performing a heart transplant than snapping together LEGO blocks. It can be done — if there’s a close enough match and the people doing it have enough skill — but the pieces don’t fit together trivially. And failure may not be immediately obvious; it may take a while to see signs of rejection.
Visualizing software development effort
As many of you know, I have a Twitter account @SansMouse that posts one Windows keyboard shortcut per day. I’m starting to experiment with adding Mac and Linux keyboard shortcuts as well. For Linux, I’ll stick to Ubuntu with the default GNOME window manager.
SansMouse will continue to post one Windows shortcut each weekday. I’ll add a #windows hash tag to these, and I’ll also start adding #mac and #ubuntu tags to tips that also work on these platforms. Over time I’ll add some tips specific to Mac and Ubuntu.
Ben Jaffe had a Twitter account @commandtab for Mac keyboard shortcuts. Ben has stopped posting to that account but he has said I could reuse his content, so I plan to fold his tweets into SansMouse.
Update (15 February 2011): Windows and Ubuntu shortcuts are quite similar, but Mac is too different. I’ve decided to drop Mac shortcuts.
Remapping caps lock
Patterns in Windows keyboard shortcuts
Visual Studio keyboard shortcuts
Samuel Hansen interviews David Spiegelhalter on his mathematical podcast Strongly Connected Components. From the show notes:
On today’s episode of Strongly Connected Components Samuel Hansen called up the Winton Professor for the Public Understanding of Risk, as well as Senior Scientist in the MRC Biostatistics Unit, David Spiegelhalter. They discussed the true meaning of risk, the importance of the Bayesian Method, how to get a lot of citations, and even a bit about the bookies.