New giant prime! New giant prime!
(It’s still really hard for me not to automatically capitalize “prime” after NaNo.)
Curtis Cooper at the University of Central Missouri came across the prime as part of the GIMPS computing project.
GIMPS, the Great Internet Mersenne Prime Search, is a distributed project designed to hunt for the rare Mersenne Primes, of which there have only been 47 found. Mersenne primes have the special form of 2p – 1, where p is itself a prime number.
The new colossal number has been confirmed as the 48th instance of a Mersenne Prime, which makes it super special. To prove the primality of the incredibly huge number, one of Cooper’s computers ran for 39 straight days. Other researchers then had to verify the primality.
I can’t wait to see the spazzing everyone falls into when we find the next DOUBLE Mersenne Prime. There are only four of them thus far discovered, you know.
Okay, so this is a result of my efforts to complete “Partying with the Primes: Part II” (see this blog for explanation. Or just scroll down a few days). Because I knew trying to get R to output some sort of number spiral would be quite an arduous task, I first decided to do a few more elementary visualizations of the primes. My first attempt led to today’s science blog.
Question: is there any sort of pattern to the spacing of prime numbers? That is, is there any sort of predictive sequence that demonstrates that the primes are “evenly spaced” (or not) amongst the other numbers?
I’d done a little bit of research on this topic prior to today, due to my 2009 NaNo (haha, we keep coming back to that, don’t we?), but it had been awhile, so I did a little bit more reading and came up with a few good sources to check out: here, here, and here.
Specifically, Zagier’s comment stood out to me: “there are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.”
So what’s a good way to visualize this stuff? My first attempt involved coding all prime numbers as “1” and all non-prime numbers as “0” and then plotting the results with 0 and 1 on the y-axis and the actual numbers (1 through whatever the highest number I chose was…I think it was 1,000), but that was a horrible mess of jagged lines and insanity, so I scrapped that and tried to think of a better way of looking at it.
In the end, I decided the best way to examine the instances of prime numbers amongst the non-primes was to plot the numbers by the numbers themselves. That is, for a given sequence of numbers (say, 1 through 10, just to make the explanation simpler) I would repeat each number by that number itself, create a new vector containing these numbers, and then plot the result.
Defunct code for better understanding:
This function says that for any number j in a given set of numbers (again, let’s say 1:10), output that number j times. So if I had the number 7, this function would give me a vector [7 7 7 7 7 7 7]‘, or 7 repeated seven times. And if I ran it for all numbers 1 through 10, I’d get the vector
[1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10]‘.
Of course, I couldn’t get this function to work but after screwing around a little bit more I finally figured out how to get this to work for larger sets of numbers, including sets just containing primes.
But what would plotting vectors like this reveal about any prevalence patterns for the primes? Well, let’s look at the plot for all numbers, shall we?
This plot is for all numbers from 1 to 1,000.
It’s pretty! Nice and smooth. So this can be said to be a plot for numbers that have a uniform or consistent pattern (all instances in this case occur one number apart, just because there’s one number difference between each instance; such is the nature of just listing the numbers 1 through 1,000).
Okay, that’s cool. So how about we look at a case where instances occur more “randomly?” In this case, I took a list of the numbers 1 through 1,000 and then went through and haphazardly deleted single numbers or large chunks of numbers so that I was left with a list that appeared to have numbers omitted at random.
Much choppier, eh? This can be said, then, to be a plot pattern for numbers that have an inconsistent or random pattern of deletion.
So what would a plot of the primes—say, all the primes below 5,000—look like?
So it’s obvious that this plot looks a lot more like the plot for numbers 1:1,000 and less like the plot involving random deletion. Interesting…I’d like to see what goes on with much larger primes, but unfortunately I can’t do that due to how huge the resulting vectors would be. R + large datasets = trouble.
So I was doing my usual surfing the internets via StumbleUpon and came across an R-Bloggers post about using R to determine the primality of any given number. The code for doing so is somewhat long and I’d like to take more time to study it and see if I could come up with my own code for determining primality, but today I was too excited to do so and instead wanted to focus on actually using the code instead.
Perhaps those readers who dig math have heard of the Ulam spiral, a method of visualizing the prime numbers in relation to the non-primes (I’m having flashback’s to NaNoWriMo 2009’s topic and therefore keep having to backspace to not capitalize “prime” and “non-prime,” haha). Developed by Stanislaw Ulam in 1963, the spiral shows a pattern indicating that certain quadratic polynomials tend to generate prime numbers. Check out the Wiki, it’s a super fascinating thing.
Anyway, ever since I’d heard of the Ulam spiral, I’ve always thought of other possible patterns or trends that may exist with respect to the primes. Could other possible patterns arise if we just “arrange” numbers in other ways? Ulam used a spiral. What other “shapes” might produce patterns?
Thus begins Part I of my mission to make pretty number patterns and see what happens! (Though I must admit that Part I is rather boring, as it just consists of me using the code on R-Bloggers).
Anyway, let’s organize this noise:
Part I: write a new function that applies R-Bloggers IsPrime() function to any given vector of numbers, say one that contains the numbers 1 through 100 (just as a start, obviously, we can extend this to much larger vectors because math rules and R is like a mental sex toy). Make sure this new function is able to output a binary response—a 0 for any non-prime and a 1 for any prime. This will allow for easy visualization once we get to that point.
Part II: Brainstorm possible pattern ideas for numbers. Figure out how in the hell to program R to output a number spiral, among other fun shapes. Use excel cells as a means by which to make the actual visualizations.
Part III: Try not to lose sanity while attempting to bend R’s base graphics to your will in order to plot said patterns without having to resort to Excel.
Part IV: Now that the work is done, actually take a step back and see if anything came of these fun experiments.
Part V: RED BULL!
Today was Part I, so I really don’t have anything special to show you guys. But next time will be fun, I promise!
30-Day Meme – Day 28: Say something to your 15 year old self.
Dear 15-Year-Old Claudia: your high school math teacher will be a jackass, but for the love of god, TAKE ALL THE MATH YOU CAN. You’ll love yourself later for it. Don’t be like the stupid 23-year-old version of yourself who quit after Algebra II (a class she totally rocked with a C-!). Tough it out, suffer through algebra, make it through trig, and ROCK OUT CALCULUS, YOU CAN SO TOTALLY DO CALCULUS. Then take all the math you can in college. You may not see it now (in fact you don’t, you see yourself right now as an artist with no need for college…this view won’t change until you’re like 19, by the way), but math and statistics are in your future. Remember back in elementary school when it was just you and two other super nerdy guys crammed in the janitor’s closet for the “advanced math” section? Remember that? Yeah, you know you can rock math. You just need to do it, yo. PRESS ON, WAYWARD HIGH SCHOOL FRESHMAN! You may feel directionless now, but that will so totally change.
See you in a few years!
I’ve been in a math mood as of late. Here are some things that are fun.
- Create an arbitrary matrix. See if you can find the eigenvalues.
- Do stuff like this. Or just mathify food in general (Tukey sandwiches! Chocolate chi-squares! Mandelbratwurst! The ever punnalicious pumpkin pi! I could go on…). I think if we all had to Fourier transform our breakfasts in the morning our lives would be a lot cooler.
- I do interval workouts on the elliptical machine in the rec center. During the high intensity intervals, I start at zero in my head, count up to the next prime number, and count backwards to zero. Then I start again, counting up to the prime number after the first prime number, count backwards to zero, and repeat ad nauseum (like this: 0,1,0; 0,1,2,1,0; 0,1,2,3,2,1,0; 0,1,2,3,5,3,2,1,0; etc.).
- Do stuff like Zeno. Whenever you have to do something for a set period of time (like 30 minutes), divide the time in half in your head then, when you reach the halfway time, divide it in half again, and again, and again, until you’ve got like a minute left. I find this helps the time pass more quickly when I’m doing something not so enjoyable. Plus it’s fun.
- This video:
Today’s song: Rich Girl$ by Down With Webster