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Stories from SlateThe Largest Known Prime Number
http://www.slate.com/articles/health_and_science/science/2016/01/the_world_s_largest_prime_number_has_22_338_618_digits_here_s_why_you_should.html
<p>Earlier this week, BBC News reported an important mathematical finding, sharing the news under the headline “<a href="http://www.bbc.com/news/technology-35361090">Largest Known Prime Number Discovered in Missouri</a>.” That phrasing makes it sound a bit like this new prime number—it’s 2<sup>74,207,281</sup>-1, by the way—was found in the middle of some road, underneath a street lamp. That’s actually not a bad way to think about it.</p>
<p>We know about this enormous prime number thanks to <a href="http://www.mersenne.org/m49/74207281.htm">the Great Internet Mersenne Prime Search</a>. The Mersenne hunt, known as GIMPS, is a large distributed computing project in which volunteers run software to search for prime numbers. Perhaps the best-known analogue is <a href="http://setiathome.ssl.berkeley.edu/">SETI@home</a>, which searches for signs of extraterrestrial life. GIMPS has had a bit more tangible success than SETI, with 15 primes discovered so far. The shiny new prime, charmingly nicknamed M74207281, is the fourth found by University of Central Missouri mathematician Curtis Cooper using GIMPS software. This one is 22,338,618 digits long.</p>
<p>A prime number is a whole number whose only factors are 1 and itself. The numbers 2, 3, 5, and 7 are prime, but 4 is not because it can be factored as 2 x 2. (For reasons of convenience, we don’t consider 1 to be a prime.) The M in GIMPS and in M74207281 stands for <a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Mersenne.html">Marin Mersenne</a>, a 17<sup>th</sup>-century French friar who studied the numbers that bear his name. Mersenne numbers are 1 less than a power of 2. Mersenne primes, logically enough, are Mersenne numbers that are also prime. The number 3 is a Mersenne prime because it’s one less than 2<sup>2</sup>, which is 4. The next few Mersenne primes are 7, 31, and 127.</p>
<p>M74207281 is the 49<sup>th</sup> known Mersenne prime. The next largest known prime, 2<sup>57,885,161</sup>-1, is also a Mersenne prime. So is the one after that. And the next one. And the next one. All in all, the 11 largest known primes are Mersenne.</p>
<p>Why do we know about so many large Mersenne primes and so few large non-Mersenne ones? It’s not because large Mersenne primes are particularly common, and it’s not a spectacular coincidence. That brings us back to the road and the street lamp. There are several different versions of <a href="http://quoteinvestigator.com/2013/04/11/better-light/">the story</a>. A man, perhaps he’s drunk, is on his hands and knees underneath a streetlight. A kind passerby, perhaps a police officer, stops to ask what he’s doing. “I’m looking for my keys,” the man replies. “Did you lose them over here?” the officer asks. “No, I lost them down the street,” the man says, “but the light is better here.”<a></a></p>
<p>We keep finding large Mersenne primes because the light is better there.</p>
<p>First, we know that only a few Mersenne numbers are even candidates for being Mersenne primes. The exponent <em>n</em> in 2<em><sup>n</sup></em>-1 needs to be prime, so we don’t need to bother to check 2<sup>6</sup>-1, for example.<a>*</a> There are a few other <a href="http://primes.utm.edu/glossary/xpage/NewMersenneConjecture.html">technical conditions</a> that make certain prime exponents more enticing to try. Finally, there’s a particular test of primeness—<a href="https://www.youtube.com/watch?v=lEvXcTYqtKU">the Lucas–Lehmer test</a>—that can only be used on Mersenne numbers.</p>
<p>To understand why the test even exists, let’s take a detour to explore why we bother finding primes in the first place. There are infinitely many of them, so it’s not like we’re going to eventually find the biggest one. But aside from being interesting in a “math for math’s sake” kind of way, finding primes is good business. <a href="http://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.html">RSA encryption</a>, one of the standard ways to scramble data online, requires the user (perhaps your bank or Amazon) to come up with two big primes and multiply them together. Assuming the encryption is implemented correctly, the difficulty of factoring the resulting product is the only thing between hackers and your credit card number.</p>
<p>This new Mersenne prime is not going to be used for encryption any time soon. Currently we only need primes that are a few hundred digits long to keep our secrets safe, so the millions of digits in M74207281 are overkill, for now.</p>
<p>You can’t just look up a 300-digit prime in a table. (There are about 10<sup>297</sup> of them. Even if we wanted to, we physically could not write them all down.) To find large primes to use in RSA encryption, we need to test randomly generated numbers for primality. One way to do this is trial division: Divide the number by smaller numbers and see if you ever get a whole number back. For large primes, this takes way too long. Hence there are primality tests that can determine whether a number is prime without actually factoring it. The Lucas-Lehmer test is one of the best.</p>
<p>The heat death of the universe would occur before we could get even a fraction of the way through trial division of a number with 22 million digits. It only took a month, however, for a computer to use the Lucas-Lehmer test to determine that M74207281 is prime. There are no other primality tests that run nearly as quickly for arbitrary 22 million–digit numbers.</p>
<p>How many primes have we missed by looking for them mostly under the Lucas-Lehmer street lamp? We don’t know the exact answer, but the <a href="http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html">prime number theorem</a> gets us close enough. It makes sense that primes get less common as we stroll out on the number line. Fully 40 percent of one-digit numbers are prime, 22 percent of two-digit numbers are prime, and only 16 percent of three-digit numbers are. The prime number theorem, first proved in the late 1800s, quantifies that decline. It says that in general, the number of primes less than <em>n</em> tends to <em>n/ln(n)</em> as <em>n</em> increases. (Here <em>ln</em> is the <a href="https://en.wikipedia.org/wiki/Natural_logarithm">natural logarithm</a>.)</p>
<p>We can use the prime number theorem to estimate how many missing primes there are between M74207281 and the next smallest prime. We just plug 2<sup>74,207,281</sup>-1 into <em>n/ln(n) </em>and get, well, a <a href="http://www.wolframalpha.com/input/?i=%282%5E74%2C207%2C281-1%29%2F%28ln%282%5E74%2C207%2C281-1%29%29">really big number</a>. We can write it most compactly by stacking exponents: 10<sup>10<sup>7.349</sup></sup>. This number has about 22,338,610 digits, give or take a couple, so we can also write it as 10<sup>22,338,610</sup>.</p>
<p>Another visit to the prime number theorem shows there are approximately 10<sup>17,425,163</sup> primes less than the next-largest known prime. That sounds impressive until you realize 10<sup>17,425,163</sup> is less than 0.000000000001 percent of 10<sup>22,338,610</sup>.</p>
<p><a></a>Stop and think about that for a moment. There are about 10<sup>22,338,610</sup> primes less than M74207281, and approximately all of them are between it and the next-largest known prime. If you want to be charitable, you could say we have some gaps in our knowledge of prime numbers. But really, it makes more sense to say we have gaps in our lack of knowledge. The <a href="http://primes.utm.edu/lists/small/millions/">millions upon millions</a> of prime numbers we’ve already found make up approximately 0 percent of the primes that are less than M74207281. Each one is a little grain of sand, a speck that does little to cover up our overwhelming ignorance of exactly where the prime numbers live.</p>
<p><em><strong>Correction, Jan. 22, 2016: </strong>This story originally referred to a possible prime number as 2<sup>n-1</sup>. That number should have been rendered as 2<sup>n</sup>-1. (<a>Return</a>.)</em></p>Fri, 22 Jan 2016 20:54:47 GMThttp://www.slate.com/articles/health_and_science/science/2016/01/the_world_s_largest_prime_number_has_22_338_618_digits_here_s_why_you_should.htmlEvelyn Lamb2016-01-22T20:54:47ZIt’s 22,338,618 digits long. That’s incredibly exciting.Health and ScienceWhy You Should Care About a Prime Number That’s 22,338,618 Digits Long100160122018sciencemathEvelyn LambSciencehttp://www.slate.com/articles/health_and_science/science/2016/01/the_world_s_largest_prime_number_has_22_338_618_digits_here_s_why_you_should.htmlfalsefalsefalseWhy you should care about a prime number that’s 22,338,618 digits long:Introducing (2^74,207,281)-1.1519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t460449660800147041282720011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t460449660800147041282720011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t460449660800147041282720011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t46044966080014704128272001Image by Eyematrix/ThinkstockRSA encryption, one of the standard ways to scramble data online, requires the user to come up with two big primes and multiply them together.Your Life in Pi
http://www.slate.com/articles/technology/technology/2013/04/pi_meme_on_reddit_and_george_takei_your_life_really_is_encoded_in_its_digits.html
<p><em>"Pi is an infinite, nonrepeating decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time, and manner of your death, and the answers to all the great questions of the universe. Converted into a bitmap, somewhere in that infinite string of digits is a pixel-perfect representation of the first thing you saw on this earth, the last thing you will see before your life leaves you, and all the moments, momentous and mundane, that will occur between those two points.</em></p>
<p><em>All information that has ever existed or will ever exist, the DNA of every being in the universe, EVERYTHING: all contained in the ratio of a circumference and a diameter."</em></p>
<p>From what I can tell, this meme comes from Redditor <a href="http://www.reddit.com/r/AskReddit/comments/zm0ac/reddit_what_is_the_most_mindblowing_sentence_you/c65xvns">kenfoldsfive's answer</a> to the question, "<a href="http://www.reddit.com/r/AskReddit/comments/zm0ac/reddit_what_is_the_most_mindblowing_sentence_you/">Reddit, what's the most mind-blowing sentence you can think of?</a>" It's been bouncing around the Internet for several months now and got a little boost recently when George Takei's Facebook page shared it with nearly 4 million followers earlier this month. It certainly is a mind-blowing idea! Infinity is always hard for our puny finite brains to handle, and I admit that the vastness of irrational numbers blows my mind whenever I accidentally think about it for too long. We're talking about a number that encodes not only my life story, but also a version in which my fly wasn't down the first time I taught a class.</p>
<p>The only problem is, it isn't true. Or, it's probably true, but we don't know for sure, and get off my lawn. At least that's what the <em><a href="http://www.huffingtonpost.com/2013/04/12/pi-meme-misleading-mathematical-constant_n_3056299.html">Huffington Post</a></em> said last week. (On Tuesday it posted a more <a href="http://www.huffingtonpost.com/david-h-bailey/are-the-digits-of-pi-random_b_3085725.html">cheerful story</a> about pi with some interesting illustrations.) The article makes some good points, but I think it misses the forest for the trees. If you look at it the right way, pi really does have it all.</p>
<p>The sticking point is the first assertion of the meme: Does pi contain every possible finite combination of digits? All irrational numbers, including pi, have infinite, nonrepeating decimal representations, but this is not enough to ensure that they include all strings. For example, 0.1010010001 … is a perfectly acceptable irrational number, and it never even includes the digit 2.</p>
<p>In an edit to the original post, kenfoldsfive notes that the statement is true if pi is a "normal number," meaning that every finite string occurs with exactly the frequency you'd expect if the digits were random. For example, 10 percent of digits are 1s, 10 percent are 2s, and so on. (This is for numbers written in base 10. Normality can be defined for binary, hexadecimal, or any other base.) In 1909, mathematician Émile Borel proved that "almost every" real number is normal. The mathematical meaning of "almost every" is more extreme than the typical English meaning. Borel showed that there is basically a 0 percent chance that if you pick a real number truly randomly, you'll get one that isn't normal.</p>
<p>For this reason and analysis of the first few trillion digits of pi, most people who care about such things believe that pi is indeed normal. There's no reason to think it isn't, except that no one has proved it yet. There's a lot we don't know about pi. After all, we only know <a href="http://www.numberworld.org/misc_runs/pi-10t/details.html">10 trillion digits of it</a>, a mere speck in the grand scheme of things. We don't even know whether every digit appears infinitely often. Maybe there are only 10<sup>1,000</sup> 7s.</p>
<p>But the focus on whether or not pi is normal misses an interesting question: Exactly how would we translate an irrational number into a bunch of text? At its heart, the meme is saying that there is something essentially infinite about irrationality that can be used to represent everything contained in our finite world. And that's right, if you choose your "code" correctly.</p>
<p>The meme suggests ASCII, a method of rendering characters using either seven or eight binary digits. (There is a decimal version as well.) Ignoring some technical details about how ASCII is really implemented, let's pretend that every two-digit combination from 00 to 99 encodes a different letter, number, space, or punctuation mark. Then we just go through the digits of pi two at a time and get some string of symbols out. If pi is normal, your life story is in there somewhere.</p>
<p>But there's nothing essential about this method of coding. My friend David Ralston at the mathematics department of SUNY Old Westbury, told me about a different way to extract text from numbers.</p>
<p>Let's say you want to find your life story in pi. We'll assume your life story isn't going to take up more room than the Bible, around 3.5 million characters in English, according to <a href="http://www.missionalchallenge.com/2008/07/how-many-letters-in-bible.html">this guy's grandfather</a>. Close enough for me. (You can pick any upper limit to the number of characters you think you'll need; the process can handle any number.) Now you need to make a list of all possible words that are no more than 3.5 million characters long. I'd recommend writing all the one-letter words first, followed by all the two-letter words, and so on, using alphabetical order at each step. This gives you some huge but still finite list of possible words—let's say K is the number of them.</p>
<p>What we need now is a way to assign each word on this list to a unique chunk of information that appears in pi. In other words, we need to find at least K distinct "things" in the number pi. To do this, we exploit the fact that for any irrational number, there are at least K+1 distinct strings that are K digits long. For example, if you look for strings that are two digits long in the irrational number 0.101001000100001 … , you will find three strings: 10, 01, and 00. For strings of three digits, you will find at least four examples.</p>
<p>Using this fact, we look at the digits of pi in chunks of length K. To the first chunk, assign the first word on your list. To the second distinct chunk (which may overlap with the first; that's OK), assign the second word, and so on. We "only" have K words on our list, and we have at least K+1 distinct blocks of length K, so we'll run out of words before we run out of blocks. If we have more distinct blocks than we have words on our list (we will), we can just start over at the beginning of our list of words. There's nothing wrong with encoding the same word twice. That will happen anyway because some K-digit blocks will show up multiple (and even infinitely many) times.</p>
<p>This process is not as straightforward as ASCII coding, but for any given irrational number, it gives us all possible strings up to some finite length. It might feel like cheating because we've specified what we want to find. But I think it's like finding a needle in a haystack: It's not going to happen if you're not looking for needles. And there is no upper limit to how long the things on your list can be, Ralston wrote in an email. It could be, "for example, all strings whose length is not larger than the number of elementary particles in the observable universe, which is (I think) a reasonable restriction to place on 'all possible words,' and includes the binary code for a very high-resolution JPEG of that awkward moment from your senior prom."</p>
<p>We don't know for sure that pi contains all possible strings of decimal digits, but on a deeper level, the meme is right. And if it gets you pondering the mysteries of infinity, so much the better. You won't be alone. The idea that monkeys sitting at a typewriter would eventually produce the complete works of Shakespeare has been around for decades. To continue bending your mind, I recommend these variations on the theme. Jorge Luis Borges explores the darker side of the concept in his short story <a href="http://jubal.westnet.com/hyperdiscordia/library_of_babel.html">The Library of Babel</a>, and the always entertaining Vi Hart's video for Pi Day 2012 tackles the question, "<a href="http://www.youtube.com/watch?v=uXoh6vi6J5U">Are Shakespeare's Plays Encoded in Pi?</a>" Enjoy!</p>Wed, 17 Apr 2013 20:43:00 GMThttp://www.slate.com/articles/technology/technology/2013/04/pi_meme_on_reddit_and_george_takei_your_life_really_is_encoded_in_its_digits.htmlEvelyn Lamb2013-04-17T20:43:00ZEverything in your past—and future—is encoded in the digits of pi.TechnologyIs Your Entire Life History Really Encoded in the Digits of Pi?100130417011memesmathmemesmathEvelyn LambTechnologyhttp://www.slate.com/articles/technology/technology/2013/04/pi_meme_on_reddit_and_george_takei_your_life_really_is_encoded_in_its_digits.htmlfalsefalsefalseIs Your Entire Life History Really Encoded in the Digits of Pi?Is Your Entire Life History Really Encoded in the Digits of Pi?Illustration by Robert Neubecker