Tuesday, 4 June 1996 will forever be remembered as a dark day for the European Space Agency (Esa). The first flight of the crewless Ariane 5 rocket, carrying with it four very expensive scientific satellites, ended after 39 seconds in an unholy ball of smoke and fire. It’s estimated that the explosion resulted in a loss of $370m (£240m). 
 
 What happened? It wasn’t a mechanical failure or an act of sabotage. No, the launch ended in disaster thanks to a simple software bug. A computer getting its maths wrong – essentially getting overwhelmed by a number bigger than it expected. 
 
 Why is the number 2,147,483,647 important? 
 
 How is it possible that computers get befuddled by numbers in this way? It turns out such errors are answerable for a series of disasters and mishaps in recent years, destroying rockets, making space probes go missing, and sending missiles off-target. So what are these bugs, and why do they happen? 
 
 Imagine trying to represent a value of, say, 105,350 miles on an odometer that has a maximum value of 99,999. The counter would “roll over” to 00,000 and then count up to 5,350, the remaining value. This is the same species of inaccuracy that doomed the 1996 Ariane 5 launch. More technically, it’s called “integer overflow”, essentially meaning that numbers are too big to be stored in a computer system, and sometimes this can cause malfunction. 
 
 Failure to launch 
 
 A full investigation of the Ariane incident found that a process left over from software in the previous generation of rockets, Ariane 4, had captured an unexpectedly high reading for the sideways velocity of the newer, faster vehicle – and the Ariane 5 rocket’s software couldn’t handle this high figure. A self-destruct sequence was initiated. A couple of seconds later, the rocket was history, as the video below shows. 
 
 
 
 Such glitches emerge with surprising frequency. It’s suspected that the reason why Nasa lost contact with the Deep Impact space probe in 2013 was an integer limit being reached. 
 
 And just last week it was reported that Boeing 787 aircraft may suffer from a similar issue. The control unit managing the delivery of power to the plane’s engines will automatically enter a failsafe mode – and shut down the engines – if it has been left on for over 248 days. Hypothetically, the engines could suddenly halt even in mid-flight. The Federal Aviation Administration’s directive on the matter states that a counter in the control unit’s software will “overflow” after this specific period of time, causing an error. Although scant details have been released – the FAA and Boeing declined to comment for this article – some amateur observers have pointed out that 248 days (when counted in 100ths of a second) is equal to the number 2,147,483,647 – which is significant. 
 
 How so? It just so happens that 2,147,483,647 is the maximum positive value that can be stored by a “32-bit signed register”, commonly installed on many computer systems. On Ariane, by comparison, the software was using a "16-bit" space, which is much smaller and only capable of storing a maximum value of 32,767. 
 
 
 
 Puzzling limit 
 
 Numbers are infinite, so why choose such limited storage spaces for them? The answer is that computers have traditionally demanded efficiency in all things. Storage space used to be much more costly than it is today and processing larger values took longer. If you kept to certain limits, software was expected to run more smoothly. Rocket guidance systems do a lot of critical number crunching very quickly, so these overheads certainly matter. The problem with that, as the Ariane 5 proved, is that such limitations aren’t always foreseen as problematic. 
 
 “We have to recognise that in software we are always approximating reality,” explains Bill Scherlis, a software expert at Carnegie Mellon University. “There’s always an engineering trade-off between the cost of having a more precise representation and the benefit of the efficiency.” 
 
 Mathematician Douglas Arnold at the University of Minnesota includes the Ariane 5 incident on a web page entitled “Some disasters attributable to bad numerical computing”. Arnold also notes the 1991 case of a Patriot missile which failed to intercept an Iraqi Scud attack on a US Army barracks during the Gulf War. In this case, an overflow error meant that the missile defence system mis-tracked the incoming Scud projectile, which was travelling at 1.7km/s, and instead scanned an area of airspace more than 500 metres from the target. 
 
 As a result, the Scud hit the barracks, killed 28 soldiers and injured a further 98 people. The full details of the computer bug in this case are quite complicated, but software engineer Andrew Lum at the University of Sydney has posted a helpful explanation of what happened, including diagrams of the Patriot system. 
 
 Gangnam glitch 
 
 Not all rollover glitches are as destructive as these examples, but they do frequently create unexpected effects. For example, in the video game Civilization, an unanticipated bug in this vein caused the peaceful character Gandhi to become uncharacteristically hostile. When players chose a certain mode to play in, the value which defined Gandhi’s aggressiveness rolled backwards past zero to the maximum. Consequently, he would threaten players with nuclear weapons at every turn – to the great amusement of many players. 
 
 And in December, it was reported that Gangnam Style, the most popular video of all time on YouTube had “broken” the website’s view counter. The counter had apparently been programmed to only run to 2,147,483,647 – again, the maximum positive value of a 32-bit signed register. It turned into good PR for YouTube, which updated the view count storage while wallowing in worldwide coverage of the site’s most popular ever video. The new maximum is well over nine quintillion. 
 
 Scherlis notes that the previous limitation reveals the expectations of the original programmers who built YouTube. “Certainly, when YouTube’s software was first developed I think it was probably hard for any developers or designers to imagine that they would overflow [this number],” he says. 
 
 It’s often this sort of assumption, which initially may seem reasonable, that causes problems years down the line. The most talked about overflow bug in history, which many will remember, was the much-hyped Millennium Bug. Although largely considered a damp squib, the Y2K problem did cause some headaches. 
 
 With Y2K, the bug was simple. What happens when you record years by the last two digits? 1900 becomes identical to 2000. Many people realised that this would cause confusion for any computer systems storing year values in this manner. As a result, a lot of advice was published in advance to programmers so that they could update systems before or on 1 January 2000. Planes did not fall from the sky, but there were some interesting consequences. For instance, radiation detection equipment in the Japanese prefecture of Ishikawa crashed at midnight; 150 slot machines at a race track in Delaware failed; and several websites gave the new date as “1 January 19100”. 
 
 Twelve years later, in a similar incident, a 105 year-old woman in Sweden named Anna Eriksson received a letter inviting her to start pre-school because software had been designed to contact individuals born in “07”. The designers intended this for people with a birthdate in 2007, not 1907 as was the case for Ms Eriksson. An inability to correctly recognise the year even led to millions of German credit and debit cards becoming unusable on New Year’s Day in 2010 
 
 The year 2038 
 
 About 15 years ago programmer William Porquet had the idea of thinking ahead to yet another crucial date – GMT 3.14.07am on Tuesday 19 January 2038. This is the moment when the number of seconds since 1 January 1970 will exceed one of the maximum values of many computers’ date and time registers nowadays. Like the Millennium Bug, failure to prepare for this could result in computer crashes. 
 
 “It was in 1999 that I first wrote about this,” comments Porquet. “I acquired the domain name 2038.org and at first it was very tongue-in-cheek. It was almost a piece of satire, a kind of an in-joke with a lot of computer boffins who say, ‘oh yes we’ll fix that in 2037…’ But then I realised there are actually some issues with this.” 
 
 Porquet is concerned about old bits of software that nobody tends to anymore – on long-established networks, or on old hardware being used in remote parts of the world. How many of them will still be in use 23 years from now and what consequences that could have is anybody’s guess. 
 
 “A lot of computer systems,” notes Porquet, “can be caused to fail in a predictable manner. But this is failure in an unpredictable manner.” 
 
 Glitch in time 
 
 Markus Kuhn, a computer scientist at the University of Cambridge explains that time related bugs create interest partly because their consequences are unpredictable, but also because they are “not unexpected” and that people are able to speculate about what will happen when the fateful date arrives. 
 
 Kuhn thinks that the 2038 problem will be less significant than Y2K because the Millennium Bug has prepared the computer industry to make the necessary fixes. Indeed, that’s all part of William Porquet’s plan. “I hope it’s something that will take me out of semi-retirement for a very large sum of money,” he says, only half joking. 
 
 For Kuhn, the interesting time problem for computers of the moment is not an overflow glitch per se, but it is one this coming June. The year 2015 will be a second longer than 2014 thanks to a move to correct the discrepancy between astronomical time (the time on Earth based on our planet’s rotation) and atomic time (the most accurate known time-keeping method, in terms of counting seconds). While atomic time, which is to be adjusted with this year’s leap second, is mind-bogglingly precise , it is actually slightly out of sync with astronomical time because the Earth’s rotation is very gradually decelerating. Geological events such as earthquakes can cause changes in the speed of this rotation meaning that the addition of leap seconds, unlike leap years, is variable. The last one was in 2012 and crashed many computers . Fortunately, says Kuhn, we’ll hopefully be more prepared than we were in the past. 
 
 It seems like no matter what we do, certain numbers and calculations will always confuse computers, causing malfunction – or worse. “We’ve learned a lot from the Y2K experience and other similar events,” notes Scherlis. “But the reality in which we are always making approximations and having to navigate an engineering trade off? That is with us forever.” 
 
 Share this story on Facebook, Google+ or Twitter. ||||| Natural number 
 
 The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes.[1] 
 
 The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772.[2] Euler used trial division, improving on Pietro Cataldi's method, so that at most 372 divisions were needed.[3] It thus improved upon the previous record-holding prime, 6,700,417, also discovered by Euler, forty years earlier. The number 2,147,483,647 remained the largest known prime until 1867.[4] 
 
 Barlow's prediction [ edit ] 
 
 At the time of its discovery, 2,147,483,647 was the largest known prime number. In 1811, Peter Barlow, not anticipating future interest in prime numbers, wrote (in An Elementary Investigation of the Theory of Numbers): 
 
 Euler ascertained that 231 − 1 = 2147483647 is a prime number; and this is the greatest at present known to be such, and, consequently, the last of the above perfect numbers [i.e., 230(231 − 1)], which depends upon this, is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for as they are merely curious, without being useful, it is not likely that any person will attempt to find one beyond it.[5] 
 
 He repeated this prediction in his 1814 work A New Mathematical and Philosophical Dictionary.[6][7] 
 
 In fact a larger prime was discovered in 1855 by Thomas Clausen (67,280,421,310,721), though a proof was not provided. Furthermore, 3,203,431,780,337 was proven to be prime in 1867.[4] 
 
 In computing [ edit ] 
 
 The number 2,147,483,647 (or hexadecimal 7FFF,FFFF 16 ) is the maximum positive value for a 32-bit signed binary integer in computing. It is therefore the maximum value for variables declared as integers (e.g., as int ) in many programming languages, and the maximum possible score, money, etc. for many video games. The appearance of the number often reflects an error, overflow condition, or missing value.[8] In December 2014, Google said that PSY's music video "Gangnam Style" had exceeded the 32-bit integer limit for YouTube view count, necessitating YouTube to upgrade the counter to a 64-bit integer.[9][10] 
 
 The data type time_t, used on operating systems such as Unix, is a signed integer counting the number of seconds since the start of the Unix epoch (midnight UTC of 1 January 1970), and is often implemented as a 32-bit integer.[11] The latest time that can be represented in this form is 03:14:07 UTC on Tuesday, 19 January 2038 (corresponding to 2,147,483,647 seconds since the start of the epoch). This means that systems using a 32-bit time_t type are susceptible to the Year 2038 problem.[12] 
 
 See also [ edit ] ||||| Earlier this week, a brief but curious post appeared on YouTube's Google+ account: "'Gangnam Style' has been viewed so many times we have to upgrade!" it said. The bizarre music video, in which a thirtysomething Korean rapper pretends to be a cowboy in a variety of inconvenient places, was an explosive worldwide smash when it landed in July 2012—but even though traffic leveled off after its viral phase, the views still continued to pile up slowly but steadily until it hit a critical 2,147,483,647 views. 
 
 So why was this particular number a problem? 
 
 Whenever a piece of data is created inside a computer, it must occupy a portion of the computer's available RAM. This is true even for something as simple as an integer, which will start to require more and more memory bits as the number gets higher. The largest number that can be represented by a 32-bit integer is 2,147,483,647. (That's 2^32, cut in half to allow an equal amount of negative integers, minus one spot which is used to represent 0.) That's more than sufficient for the majority of routine calculations—but it's not enough to handle the online populace's appetite for Psy. 
 
 In cases where even larger numbers are needed—calculations involving the many trillions of dollars of the U.S. GDP, for example, or astronomical measurements—it's possible to use 64-bit integers instead. The memory footprint for each of these data points will be twice as heavy, but they can then be used to store far higher numbers. So what's the new 64-bit limit? 9,223,372,036,854,775,808. And that, for those of you keeping track, is the new view count Psy's music video will have to hit to outpace YouTube's ability to track it a second time. 
 
 When "Gangnam Style" hit 2 billion views back in May, a record at the time, Google's engineers noted that it might also soon overflow the limits of a 32-bit integer. That fix was as simple as changing the memory space allotted to that data point from 32 bit to 64 bit, which is actually a relatively small change. It didn't actually "break YouTube's code" in any meaningful sense, let alone "break the Internet." 
 
 What's mind-boggling about all this, though, is that the number that was pushing those limits loosely correlates with a quantity of people. It's the play count, strictly speaking, but music video views don't overflow theoretical mathematical boundaries unless a lot—like, a lot—of people are watching them. That number may be only implied, but it's also huge. Thus, YouTube's recent change to the view count limit is interesting not as a computational problem, nor as an example of a meme spinning out of control, but rather because we've just seen a population of active technology users overflowing the previously accepted limits of mathematics in computing; OS X only moved beyond 32-bit processors with version 10.7, which was released just one year before "Gangnam Style." It's not the height of the aggregate number that's compelling, so much as the breadth of the user base that necessitated it. 
 
 All these people were interacting with a single technical platform—with a single item presented on that platform, even. In comparison, with around 1.35 billion users, Facebook still has plenty of headroom to continue to represent their User ID numbers with a 32-bit data point. But what happens as the world's population continues to grow, and as previously dark parts of China and India start to come online permanently? The number of people in the world already exceeds the limits of 32-bit integers, and in the foreseeable future, a majority of them are probably going to possess some combination of phones, laptops, tablets, and wearable computing devices. 
 
 Already, the term "Big Data" has become a buzzword in the tech industry, denoting the study of data sets that are too large to be processed using conventional techniques. But maybe it's those conventions that need to evolve, now that the number of people who substantively engage with technology is no longer a 32-bit matter. It's probably time to buy some extra RAM, but feel free pause first to swoon at the growing scope of the world if you need to. ||||| Crawl of outlinks from wikipedia.org started March, 2016. These files are currently not publicly accessible. Properties of this collection. It has been several years since the last time we did this. For this collection, several things were done: 1. Turned off duplicate detection. This collection will be complete, as there is a good chance we will share the data, and sharing data with pointers to random other collections, is a complex problem. 2. For the first time, did all the different wikis. The original runs were just against the enwiki. This one, the seed list was built from all 865 collections.
– As complex and incredible as our latest gadgets are, they can be foiled by something very simple: a number that's just too big. The BBC explains the phenomenon in terms of an odometer that only goes up to 99,999 miles. Drive another mile, and it would roll to zero again, and start counting from there. What is known as "integer overflow" can, in some cases, prompt disaster, the BBC notes. Case in point: a European Space Agency rocket that exploded shortly after it launched because velocity readings were beyond its capacity. In other cases, the result is comical. Take, for instance, the 105-year-old woman who got an invitation to preschool because she was born in '07—though not 2007, as the software involved expected. One number in particular is a problem for many of today's systems: 2,147,483,647, a number with its own Wikipedia page. It's the highest number that a "32-bit signed register," as many systems run, can handle; the limit exists thanks to a need for digital efficiency, an expert tells the BBC. YouTube saw the problem firsthand: "We never thought a video would be watched in numbers greater than a 32-bit integer (=2,147,483,647 views), but that was before we met PSY," the site noted, referring to the musician whose "Gangnam Style" video "broke" the counter. (As Wired explained in 2014, YouTube figured out a fix by going from 32-bit integers to 64-bit, so the view limit is now in the quintillions.) More worryingly, the Boeing 787 Dreamliner may encounter a problem if its equipment has been left running for 2,147,483,647 hundredths of a second, or 248 days. And there are fears comparable to those surrounding the Y2K bug centered on the year 2038. Fortunately, experts are on the lookout for such issues. (Click to see how a "leap second" brought down a chunk of the Web.)