Archive for September 27th, 2008|Daily archive page

Some really big prime number just got found

Mathematicians at UCLA have just discovered a prime number with almost 13 million digits. Prime numbers, of course, are only divisible by themselves and one; it has been known since Euclid in the third century BCE that there are an infinite number of prime numbers.

However, this is a special type of prime number called a Mersenne Prime. Named for 17th century French mathematician Marin Mersenne, these numbers are simply one less than a power of two (2^n – 1). Only 47 have been discovered to date, all but ten of them first identified since the start of the 20th century. Incidentally, the largest currently known prime number happens to be a Mersenne Prime: 243,112,609 − 1.

Math people get really excited over these numbers and when new ones are discovered. I don’t see why. There is no practical use for these numbers. What’s the big deal? I could just as easily make up the “Jacob Prime” which, uh, is… three four less than a power of two. Wow; how special is that? Yeah, okay, it sucks—but that’s my point.

But I don’t dislike prime numbers. One thing about them that I do find very interesting is the Ulam spiral phenomenon, named after its discoverer, Polish mathematicial Stanislaw Ulam. He stumbled upon them while at doodling at a really boring meeting (probably where they were discussing Mersenne Primes) and you can produce the phenomenon this way: (1) write down all integers starting at one point and spiraling outward (see firgure 1 below) and then (2) either circling, as Ulam did, all the primes, or, as below, removing all the non primes (see figure 2).

See paragraph above for explanation

Start by writing all integers in a spiral, then remove all the numbers that aren't prime to find something quite interesting...

Once you have done enough numbers—and it doesn’t take many—you’ll see that the prime numbers tend to occur along orthogonal lines. See, for instance, the lines formed by 3-13-31, 41-19-5, and 19-7-23-47 in figure 2.

The Ulam spiral phenomenon is clearly seen here on a 200 x 200 grid. Note the orthogonal lines throughout the image.

The Ulam spiral phenomenon is clearly seen here on a 200 x 200 grid. Note the diagonal lines throughout the image.

All prime numbers, except for 2, are odd numbers; and since in the Ulam spiral adjacent diagonals are alternatively odd and even numbers, it is not surprising that all prime numbers lie in alternate diagonals. However, what is surprising is the tendency of prime numbers to lie on some diagonals more than others; there is no apparent reason for this to be. This tendency occurs on any scale and regardless of what integer you start with at the middle. At right is a 200 x 200 grid of numbers and the effect is clearly seen. I am furthermore told that, at sufficient distances from the center, horizontal and vertical lines also become evident.

This effect is built into the nature of numbers, it’s not something that some guy just made up. Admittedly, it’s no more useful than Mersenne primes, but it’s a whole lot cooler and doesn’t take huge amounts of computing power to play around with—just grab a piece of paper and a pen!

Anyway, those guys who found the 13 million digit Mersenne prime are going to win $100,000 (I told you, mathematicians go nuts over these things). I’ll give a nickel to whoever finds the first Jacob prime with more than 1207 digits. Start your calculators!

China performs space walk, rips off Star Trek

The logo of the China National Space Administration looks strangely familiar

The logo of the China National Space Administration looks strangely familiar

Congratulations to the People’s Republic of China, which has launched their third crewed mission to space and Zhai Zhigang, age 41, has successfully performed his nation’s first space walk. The spacecraft, the Shenzhou 7, has a crew of three astronauts—or taikonauts, as the Chinese call them (cf. cosmonaut). Besides China, only the United States and the Soviet Union/Russia have launched people into space.

China hopes to establish a space station by 2020 and also has plans to land people on the Moon and, eventually, Mars. They made the news last year when they shot down an old satellite in a test of their military abilities; this was largely seen as a provocative act and a possible threat to the United States, which maintains considerable assets in space for both communication, intelligence, and scientific purposes.

I am also struck, however, by the unoriginality of the China National Space Administration logo. Just look at it. Doesn’t it remind you of something? If you’re a fan of Star Trek, I’ll bet that it does. Compare:

One possible origin of the CNSA logo

One possible origin of the CNSA logo

Regardless of where the logo came from, Godspeed to the three taikonauts, Zhai Zhigang, Liu Boming, and Jing Haipeng.

Intrade markets predict Obama victory

Following last night’s first presidential debate, speculators playing the Intrade prediction markets still anticipate an Obama victory on November 4th.  Intrade is a gambling site where you can bet real money that a certain event will or won’t happen, and the presidential election is just one such event; you buy shares that will pay a fixed amount, I believe $10, if the event happens and nothing if it doesn’t; the value of the shares varies as people buy and sell them as their perception of the likelihood of the event occurring rises and falls.

At this moment, shares that will pay $10 if Obama wins (and nothing if he doesn’t) are trading at $5.60 and shares that will pay $10 if McCain wins (and nothing if he doesn’t) are trading at $4.36.  This means that gamblers—or investors, if you prefer—think there is about a 56% chance Obama will win and about a 43.6% chance that McCain will win; this is a slight move in Obama’s favor since last night’s debate.  They also sell shares for the contests in each of the 50 states plus the district of Columbia.  Currently, speculators think that Obama will win 311 electoral votes and McCain will gain 227, this is with the Democrat winning Nevada, Colorado, Virginia, and, by the thinnest margins, Ohio.

These predictions markets are somewhat like polls in how they attempt to gauge support and predict the ultimate outcome.  However, unlike polls, they anticipate future moves: investors knew that Obama’s poll numbers would rise during the Democratic convention and McCain’s during the Republican convention and share prices took this into account and didn’t move when the expected bumps came.

People interested in the outcome of the election, which is hopefully everyone, may find it interesting to watch the political futures markets.  By putting real money at stake, they attempt to harness the wisdom of crowds, which is often better at predicting the future than even what experts say.

I recently saw the wisdom of crowds at work, though in a considerably different context.  One gentleman, a fireman, had shared an anecdote about the aluminum siding on a house melting in the course of one conflagration he witnessed.  This got me to asking what the melting point of Aluminum is; I didn’t know, and neither did anyone else.  Eventually, I took guesses from everyone to see who could come the closest, promising a prize to the winner.  There were ten guesses, ranging from 162.5° F all the way up to 3200° F.  The actual melting point of Aluminum is 1220.6° F; the two closes guesses were 550° and 1800°.  However, if we’d taken the average of all the guesses (omitting the guess of 8° by the two year old son of one attendee) we’d have 1031.95°—which would have been, by far, the most accurate and would have won the prize: a nickel.

In any event, if you find poll numbers interesting, check out the Intrade prediction market, it just may prove more accurate than Gallup.