Archive for the ‘Stanislaw Ulam’ Tag

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!