Largest Prime Number Discovered

Back when I was in college, I participated in the great GIMPS Project, searching for what is known for a Mersenne prime number (Mersenne primes are of the form (2^X)-1, with the first primes being 3, 7, 31, and 127 corresponding to X = 2, 3, 5, and 7, respectively). My computer would use its extraneous resources to help in the search, and while nothing ever came of it, it’s pretty cool to know that I made a modest contribution to the project. So it was great to learn today that the GIMPS Project found the largest prime number ever as of January 2013. The largest (known) prime number now is 2^57,885,161-1, and its discovery is noted on this post:

The new prime number is a member of a special class of extremely rare prime numbers known as Mersenne primes. It is only the 48th known Mersenne prime ever discovered, each increasingly difficult to find. Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers more than 350 years ago. GIMPS, founded in 1996, has discovered all 14 of the largest known Mersenne primes. Volunteers download a free program to search for these primes with a cash award offered to anyone lucky enough to compute a new prime. Chris Caldwell maintains an authoritative web site on the largest known primes as well as the history of Mersenne primes.

To prove there were no errors in the prime discovery process, the new prime was independently verified using different programs running on different hardware. Serge Batalov ran Ernst Mayer’s MLucas software on a 32-core server in 6 days (resource donated by Novartis[2] IT group) to verify the new prime. Jerry Hallett verified the prime using the CUDALucas software running on a NVidia GPU in 3.6 days. Finally, Dr. Jeff Gilchrist verified the find using the GIMPS software on an Intel i7 CPU in 4.5 days and the CUDALucas program on a NVidia GTX 560 Ti in 7.7 days.

This largest prime number contains 17,425,170 digits. If you have a fast Internet connection, you can see how huge this number is (with all of its digits written out one by one) by clicking here. Pretty cool.

Felix Baumgartner: The Mathematics of Falling Faster than the Speed of Sound

Earlier this month, Felix Baumgartner jumped from 39,045 meters, or 24.26 miles, above the Earth from a capsule lifted by a 334-foot-tall helium filled balloon (twice the height of Nelson’s column and 2.5 times the diameter of the Hindenberg). The jump was equivalent to a fall from 4.4 Mount Everests stacked on top of each other, or falling 93% of the length of a marathon.

At 24.26 miles above the Earth, the atmosphere is very thin and cold, only about -14 degrees Fahrenheit on average. The temperature, unlike air pressure, does not change linearly with altitude at such heights. Jason Martinez, a programmer at Wolfram|Alpha, did a number of calculations to see how Felix’s record-setting jump came to be. It’s a very math-heavy post, but something I have been looking forward to!

 

Felix Baumgartner’s mach speed at various portions of his free fall.

If you’re into the nitty-gritty mathematical computations, I suggest reading the whole blog post.

Fractal Kitties: They Exist

Fun post at Scientific American explaining how fractal kitties can explain Julia sets:

Julia sets for polynomials of degree two are well-understood, although they’re often fractals rather than simple shapes such as circles. The story gets a lot more complicated as the degree increases because higher-degree polynomials are difficult to factor. (The much-maligned quadratic formula—the reason why we can easily discern the roots of degree two polynomials—is our friend!) A little bit is known about the possible shapes for Julia sets of degree 3 and 4 polynomials, but the shapes of the Julia sets of arbitrary polynomials are not yet understood.

Lindsey is a graduate student in mathematics at Cornell University. Her advisor is John Smillie, but Thurston was an unofficial second advisor, and it was his idea to start this research project. “I was sitting in his house, and he was staring off into space and asked, ‘I wonder if Julia sets can be made into shapes,’” she says. Thurston had been working on understanding the Mandelbrot set better, and looking at the shapes of Julia sets was a related pursuit. The Mandelbrot set, one of the most famous fractals, is closely related to Julia sets of degree two polynomials: imagine the polynomial z2+c, where c can be any complex number. The number c is in the Mandelbrot set if 0 is in the filled Julia set of z2+c.

Fractal Kitty!

The math may get hairy at times…but then again, so do the images. Ha!

The Truth about the Golden Ratio

Steven Strogatz, in a piece titled “Proportion Control” at The New York Times, dispels the myths behind the famous number known as the golden ratio (φequal to 1/2+√5/2):

Unfortunately, in the more than two millenniums since Euclid, the golden ratio has suffered from so much hype, numerology and wishful thinking that it’s become hard to separate the myth from the math. Many of its supposed occurrences in nature, anatomy, art and architecture don’t stand up to careful scrutiny. For example, you can find lots of books and Web sites claiming that the shell of the chambered nautilus obeys the golden ratio, but in reality, nautilus shells have average growth ratios between 1.24 and 1.43, quite far from 1.618.

So be skeptical the next time you see the golden ratio being used to sell blue jeans, stock tips or the perfect smile.

The upside is, if a nautilus can’t get its proportions golden, maybe I shouldn’t worry so much about mine.

Pass the nachos.

Love that conclusion.

Physicist Uses Math, Writes Paper, To Beat Traffic Ticket

What would you do to get out of a traffic ticket, if you were convinced you were innocent? Probably not as much as Dmitri Krioukov, a physicist based at the University of California San Diego, who was fined for (purportedly) running a stop sign. In a paper titled “Proof of Innocence,” Krioukov argues three physical phenomena combined at just the right time and misled the officer:

We show that if a car stops at a stop sign, an observer, e.g., a police ocer, located at a certain distance perpendicular to the car trajectory, must have an illusion that the car does not stop, if the following three conditions are satis ed: (1) the observer measures not the linear but angular speed of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at the moment when both cars are near the stop sign.

When Krioukov drove toward the stop sign the police officer was approximating Krioukov’s angular velocity instead of his linear velocity. This happens when we try to estimate the speed of a passing object, and the effect is more pronounced for faster objects. In Krioukov’s case, the police cruiser was situated about 100 feet away from a perpendicular intersection with a stop sign. Consequently, a car approaching the intersection with constant linear velocity will rapidly increase in angular velocity from the police officer’s perspective. A sneeze caused Krioukov to slam on the brakes hard as he approached the stop sign. With a potential car blocking the officer’s view for a split second, it appeared as though Krioukov never slowed down.

This mathematical description swayed the judge (or maybe he was simply impressed by Krioukov dedication in writing a paper on this personal incident), and the case was dismissed. What a way to get out of paying a traffic ticket!

###

(via Physics Central)

Finding Waldo with Mathematica

For those of you who are fans of Finding Waldo and have a bit of a nerdy side to you, you’ll appreciate that someone figured out how to find Waldo using Mathematica:

Finding Waldo with the help of Mathematica.

The author describes his technique and provides the relevant code:

First, I’m filtering out all colours that aren’t red

waldo = Import["http://www.findwaldo.com/fankit/graphics/IntlManOfLiterature/Scenes/DepartmentStore.jpg"]; 
red = Fold[ImageSubtract, #[[1]], Rest[#]] &@ColorSeparate[waldo]; 

Next, I’m calculating the correlation of this image with a simple black and white pattern to find the red and white transitions in the shirt.

corr = ImageCorrelate[red,    Image@Join[ConstantArray[1, {2, 4}], 
ConstantArray[0, {2, 4}]],    NormalizedSquaredEuclideanDistance]; 

I use Binarize to pick out the pixels in the image with a sufficiently high correlation

and draw white circle around them to emphasize them using Dilation

pos = Dilation[ColorNegate[Binarize[corr, .12]], DiskMatrix[30]]; 

I had to play around a little with the level. If the level is too high, too many false positives are picked out.

Finally I’m combining this result with the original image to get the result above

found = ImageMultiply[waldo, ImageAdd[ColorConvert[pos, "GrayLevel"], .5]]

Amazing.

###

(via Kottke)

The Charlemagne Riddle and Pedigree Collapse

Robert Krulwich recounts a story/riddle about a guy who discovered he was related to Charlemagne the Great. But was he? He went to class one day, where the teacher had a lesson on genealogy:

The teacher says if you count your direct ancestors backward through time, the further back you go, obviously, the more ancestors you have. But when you do the numbers, something queer happens.

Go back to A.D. 800, he said, and the number of direct ancestors is, well, puzzling. You start with two grandparents, then four great-grandparents, then on to eight, 16, etc., and by the time you get to A.D. 800, the number averages to about 562,949,953,421,321. That’s a lot of people. In fact, that’s more people than have ever lived.

So if you go far back enough in history, everyone is related to Charlemagne. This answer is justified with the so-called pedigree collapse:

Without pedigree collapse, a person’s ancestor tree is a binary tree, formed by the person, the parents (2), the grandparents (4), great-grandparents (8), and so on. However, the number of individuals in such a tree grows exponentially and will eventually become impossibly high. For example, a single individual alive today would, over 30 generations going back to the High Middle Ages, have 230 or roughly a billion ancestors, more than the total world population at the time.

This apparent paradox is explained by shared ancestors. Instead of consisting of all unique individuals, a single individual may occupy multiple places in the tree. This typically happens when the parents of an ancestor are cousins (sometimes unbeknownst to themselves). For example, the offspring of two first cousins has at most only six great-grandparents instead of the normal eight. This reduction in the number of ancestors is pedigree collapse. It collapses the ancestor tree into a directed acyclic graph.

In some cultures, cousins were encouraged or required to marry to keep kin bonds, wealth and property within a family (endogamy). Among royalty, the frequent requirement to only marry other royals resulted in a reduced gene pool in which most individuals were the result of extensive pedigree collapse

So next time someone says they’re a direct descendant of someone, say, who was born before 1600, be highly, highly skeptical.