tag:theconversation.com,2011:/africa/topics/prime-numbers-17600/articlesPrime numbers – The Conversation2018-09-27T15:32:38Ztag:theconversation.com,2011:article/1039742018-09-27T15:32:38Z2018-09-27T15:32:38ZHas one of math's greatest mysteries, the Riemann hypothesis, finally been solved?<figure><img src="https://images.theconversation.com/files/238231/original/file-20180926-48665-q7vcjw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A prime mystery.</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/prime-152874113?src=ywpcul8puqjSR_z4VssLpw-1-2">Robert Lessmann/shutterstock.com</a></span></figcaption></figure><p>Over the past few days, the mathematics world has been abuzz over the news that Sir Michael Atiyah, the famous Fields Medalist and Abel Prize winner, <a href="https://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis">claims to have solved the Riemann hypothesis</a>. </p>
<p>If his proof turns out to be correct, this would be one of the most important mathematical achievements in many years. In fact, this would be one of the biggest results in mathematics, comparable to the proof of <a href="https://www.scientificamerican.com/article/are-mathematicians-finall/">Fermat’s Last Theorem from 1994</a> and the proof of the <a href="https://www.aps.org/publications/apsnews/201311/physicshistory.cfm">Poincare Conjecture from 2002</a>. </p>
<p>Besides being one of the great unsolved problems in mathematics and therefore garnishing glory for the person who solves it, the Riemann hypothesis is one of the Clay Mathematics Institute’s <a href="http://www.claymath.org/millennium-problems">“Million Dollar Problems.”</a> A solution would certainly yield a pretty profitable haul: one million dollars. </p>
<p><a href="https://www.britannica.com/science/Riemann-hypothesis">The Riemann hypothesis</a> has to do with the distribution of the prime numbers, those integers that can be divided only by themselves and one, like 3, 5, 7, 11 and so on. We know from the Greeks that there are infinitely many primes. What we don’t know is how they are distributed within the integers. </p>
<p>The problem originated in estimating the so-called “prime pi” function, an equation to find the number of primes less than a given number. But its modern reformulation, by German mathematician Bernhard Riemann in 1858, has to do with the location of the zeros of what is now known as the Riemann zeta function. </p>
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<a href="https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=2818&fit=crop&dpr=1 600w, https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=2818&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=2818&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=3541&fit=crop&dpr=1 754w, https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=3541&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/238356/original/file-20180927-48656-1i3kapj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=3541&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">A visualization of the Riemann zeta function.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Thumbnail-of-Riemann_Zeta.png">Jan Homann/Wikimedia</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>The technical statement of the Riemann hypothesis is “the zeros of the Riemann zeta function which lie in the critical strip must lie on the critical line.” Even understanding that statement involves graduate-level mathematics courses in complex analysis. </p>
<p>Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much. Only an abstract proof will do. </p>
<p>If, in fact, the Riemann hypothesis were not true, then mathematicians’ current thinking about the distribution of the prime numbers would be way off, and we would need to seriously rethink the primes. </p>
<p>The Riemann hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the sort of problem that an inexperienced math student can play around with in his or her spare time. Attempts at verifying it involve many very deep tools from complex analysis and are usually very serious ones done by some of the best names in mathematics. </p>
<p>Atiyah gave a lecture in Germany on Sept. 25 in which he presented an outline of his approach to verify the Riemann hypothesis. This outline is often the first announcement of the solution but should not be taken that the problem has been solved – far from it. For mathematicians like me, the “proof is in the pudding,” and there are many steps that need to be taken before the community will pronounce Atiyah’s solution as correct. First, he will have to circulate a manuscript detailing his solution. Then, there is the painstaking task of verifying his proof. This could take quite a lot of time, maybe months or even years. </p>
<p>Is Atiyah’s attempt at the Riemann hypothesis serious? Perhaps. His reputation is stellar, and he is certainly capable enough to pull it off. On the other hand, there have been several other serious attempts at this problem that did not pan out. At some point, Atiyah will need to circulate a manuscript that experts can check with a fine-tooth comb.</p><img src="https://counter.theconversation.com/content/103974/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>William Ross does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>A famous mathematician claims to have cracked a longstanding problem involving prime numbers. But his results still need to be verified.William Ross, Professor of Mathematics, University of RichmondLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/924842018-04-02T10:47:33Z2018-04-02T10:47:33ZWhy prime numbers still fascinate mathematicians, 2,300 years later<figure><img src="https://images.theconversation.com/files/211771/original/file-20180323-54878-15xsrf7.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Primes still have the power to surprise. </span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/prime-number-collage-437862145?src=ywpcul8puqjSR_z4VssLpw-1-0">Chris-LiveLoveClick/shutterstock.com</a></span></figcaption></figure><p>On March 20, American-Canadian mathematician Robert Langlands <a href="http://www.abelprize.no/nyheter/vis.html">received the Abel Prize</a>, celebrating lifetime achievement in mathematics. Langlands’ research demonstrated how concepts from geometry, algebra and analysis could be brought together by a common link to prime numbers. </p>
<p>When the King of Norway presents the award to Langlands in May, he will honor the latest in a 2,300-year effort to understand prime numbers, arguably the biggest and oldest data set in mathematics. </p>
<p>As a mathematician devoted to this <a href="https://www.quantamagazine.org/robert-langlands-mathematical-visionary-wins-the-abel-prize-20180320/">“Langlands program,”</a> I’m fascinated by the history of prime numbers and how recent advances tease out their secrets. Why they have captivated mathematicians for millennia?</p>
<h2>How to find primes</h2>
<p>To study primes, mathematicians strain whole numbers through one virtual mesh after another until only primes remain. This sieving process produced tables of millions of primes in the 1800s. It allows today’s computers to <a href="https://primesieve.org/">find billions of primes in less than a second</a>. But the core idea of the sieve has not changed in over 2,000 years.</p>
<p>“A prime number is that which is measured by the unit alone,” <a href="http://data.perseus.org/citations/urn:cts:greekLit:tlg1799.tlg001.perseus-eng1:7.def.11">mathematician Euclid wrote in 300 B.C.</a> This means that prime numbers can’t be evenly divided by any smaller number except 1. By convention, mathematicians don’t count 1 itself as a prime number. </p>
<p>Euclid proved the infinitude of primes – they go on forever – but history suggests <a href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015005675411;view=1up;seq=220">it was Eratosthenes</a> who gave us the sieve to quickly list the primes. </p>
<p>Here’s the idea of the sieve. First, filter out multiples of 2, then 3, then 5, then 7 – the first four primes. If you do this with all numbers from 2 to 100, only prime numbers will remain.</p>
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<img width="100%" src="https://media.giphy.com/media/1ppTHkroIGmfybV0x9/giphy.gif">
<figcaption><span class="caption">Sieving multiples of 2, 3, 5 and 7 leaves only the primes between 1 and 100. Courtesy of M.H. Weissman.</span></figcaption>
</figure>
<p>With eight filtering steps, one can isolate the primes up to 400. With 168 filtering steps, one can isolate the primes up to 1 million. That’s the power of the sieve of Eratosthenes.</p>
<h2>Tables and tables</h2>
<p>An early figure in <a href="http://eudml.org/doc/275002">tabulating primes</a> is John Pell, an English mathematician who dedicated himself to creating tables of useful numbers. He was motivated to solve ancient arithmetic problems of Diophantos, but also by a personal quest to organize mathematical truths. Thanks to his efforts, the primes up to 100,000 were widely circulated by the early 1700s. By 1800, independent projects had tabulated the primes up to 1 million. </p>
<p>To automate the tedious sieving steps, a German mathematician named Carl Friedrich Hindenburg used adjustable sliders to stamp out multiples across a whole page of a table at once. Another low-tech but effective approach used stencils to locate the multiples. By the mid-1800s, mathematician <a href="http://locomat.loria.fr/kulik1825/kulik1825doc.pdf">Jakob Kulik</a> had embarked on an ambitious project to find all the primes up to 100 million.</p>
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<figcaption>
<span class="caption">A stencil used by Kulik to sieve the multiples of 37.</span>
<span class="attribution"><span class="source">AÖAW, Nachlass Kulik, Image courtesy of Denis Roegel</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>This “big data” of the 1800s might have only served as reference table, if <a href="https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss">Carl Friedrich Gauss</a> hadn’t decided to analyze the primes for their own sake. Armed with a list of primes up to 3 million, Gauss began counting them, one “<a href="https://doi.org/10.1090/S0273-0979-05-01096-7">chiliad</a>,” or group of 1000 units, at a time. He counted the primes up to 1,000, then the primes between 1,000 and 2,000, then between 2,000 and 3,000 and so on. </p>
<p>Gauss discovered that, as he counted higher, the primes gradually become less frequent according to an “inverse-log” law. Gauss’s law doesn’t show exactly how many primes there are, but it gives a pretty good estimate. For example, his law predicts 72 primes between 1,000,000 and 1,001,000. The <a href="https://primes.utm.edu/lists/small/100000.txt">correct count is 75 primes</a>, about a 4 percent error. </p>
<p>A century after Gauss’ first explorations, his law was proved in the <a href="http://www.jstor.org/stable/2319162">“prime number theorem.”</a> The percent error approaches zero at bigger and bigger ranges of primes. The Riemann hypothesis, <a href="http://www.claymath.org/millennium-problems/riemann-hypothesis">a million-dollar prize problem</a> today, also describes how accurate Gauss’ estimate really is. </p>
<p>The prime number theorem and Riemann hypothesis get the attention and the money, but both followed up on earlier, less glamorous data analysis.</p>
<h2>Modern prime mysteries</h2>
<p>Today, our data sets come from computer programs rather than hand-cut stencils, but mathematicians are still finding new patterns in primes.</p>
<p>Except for 2 and 5, all prime numbers end in the digit 1, 3, 7 or 9. In the 1800s, it was proven that these possible last digits are equally frequent. In other words, if you look at the primes up to a million, about 25 percent end in 1, 25 percent end in 3, 25 percent end in 7, and 25 percent end in 9.</p>
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<p>A few years ago, Stanford number theorists Robert Lemke Oliver and Kannan Soundararajan were caught off guard by quirks in the final digits of primes. <a href="https://www.quantamagazine.org/mathematicians-discover-prime-conspiracy-20160313/">An experiment</a> looked at the last digit of a prime, as well as the last digit of the very next prime. For example, the next prime after 23 is 29: One sees a 3 and then a 9 in their last digits. Does one see 3 then 9 more often than 3 then 7, among the last digits of primes? </p>
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<figcaption>
<span class="caption">Frequency of last-digit pairs, among successive prime numbers up to 100 million. Matching colors correspond to matching gaps.</span>
<span class="attribution"><span class="source">M.H. Weissman</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
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<p>Number theorists expected some variation, but what they found far exceeded expectations. Primes are separated by different gaps; for example, 23 is six numbers away from 29. But 3-then-9 primes like 23 and 29 are far more common than 7-then-3 primes, even though both come from a gap of six. </p>
<p>Mathematicians soon found a <a href="http://www.pnas.org/content/pnas/113/31/E4446.full.pdf">plausible explanation</a>. But, when it comes to the study of successive primes, mathematicians are <a href="https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/">(mostly)</a> limited to data analysis and persuasion. Proofs – mathematicians’ gold standard for explaining why things are true – seem decades away.</p><img src="https://counter.theconversation.com/content/92484/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Martin Weissman receives funding from the Simons Foundation for collaboration in mathematics. </span></em></p>Prime numbers are the biggest and oldest data set in mathematics. Why have they captivated mathematicians for millennia?Martin H. Weissman, Associate Professor of Mathematics, University of California, Santa CruzLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/898782018-01-12T12:11:59Z2018-01-12T12:11:59ZWhy do we need to know about prime numbers with millions of digits?<figure><img src="https://images.theconversation.com/files/201796/original/file-20180112-101489-x4wvpo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">SeventyFour via Shutterstock</span></span></figcaption></figure><p>Prime numbers are <a href="https://theconversation.com/your-numbers-up-a-case-for-the-usefulness-of-useless-maths-11799">more than just numbers</a> that can only be divided by themselves and one. They are a mathematical mystery, the secrets of which mathematicians have been trying to uncover ever since Euclid <a href="https://www.math.utah.edu/%7Epa/math/q2.html">proved</a> that they have no end.</p>
<p>An ongoing project – the <a href="https://www.mersenne.org">Great Internet Mersenne Prime Search</a> – which aims to discover more and more primes of a particularly rare kind, has recently resulted in the discovery of the largest prime number known to date. Stretching to 23,249,425 digits, it is so large that it would easily fill 9,000 book pages. By comparison, the number of atoms in the entire observable universe is estimated to have no more than 100 digits. </p>
<p>The number, simply written as 2⁷⁷²³²⁹¹⁷-1 (two to the power of 77,232,917, minus one) was <a href="https://theconversation.com/largest-known-prime-number-discovered-why-it-matters-89743">found by a volunteer</a> who had dedicated <a href="https://www.mersenne.org/primes/press/M77232917.html">14 years of computing time</a> to the endeavour. </p>
<p>You may be wondering, if the number stretches to more than 23m digits, why we need to know about it? Surely the most important numbers are the ones that we can use to quantify our world? That’s not the case. We need to know about the properties of different numbers so that we can not only keep developing the technology we rely on, but also keep it secure.</p>
<h2>Secrecy with prime numbers</h2>
<p>One of the most widely used applications of prime numbers in computing is <a href="https://theconversation.com/the-rsa-algorithm-or-how-to-send-private-love-letters-13191">the RSA encryption system</a>. In 1978, Ron Rivest, Adi Shamir and Leonard Adleman combined some simple, known facts about numbers to create RSA. The system they developed allows for the secure transmission of information – such as credit card numbers – online.</p>
<p>The first ingredient required for the algorithm are two large prime numbers. The larger the numbers, the safer the encryption. The counting numbers one, two, three, four, and so on – also called the natural numbers – are, obviously, extremely useful here. But the prime numbers are the building blocks of all natural numbers and so even more important. </p>
<p>Take the number 70 for example. Division shows that it is the product of two and 35. Further, 35 is the product of five and seven. So 70 is the product of three smaller numbers: two, five, and seven. This is the end of the road for 70, since none of these can be further broken down. We have found the primal components that make up 70, giving its prime factorisation. </p>
<p>Multiplying two numbers, even if very large, is perhaps tedious but a straightforward task. Finding prime factorisation, on the other hand, is extremely hard, and that is precisely what the RSA system takes advantage of. </p>
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<p>Suppose that Alice and Bob wish to communicate secretly over the internet. They require an encryption system. If they first meet in person, they can devise a method for encryption and decryption that only they will know, but if the initial communication is online, they need to first openly communicate the encryption system itself – a risky business. </p>
<p>However, if Alice chooses two large prime numbers, computes their product, and communicates this openly, finding out what her original prime numbers were will be a very difficult task, as only she knows the factors. </p>
<p>So Alice communicates her product to Bob, keeping her factors secret. Bob uses the product to encrypt his message to Alice, which can only be decrypted using the factors that she knows. If Eve is eavesdropping, she cannot decipher Bob’s message unless she acquires Alice’s factors, which were never communicated. If Eve tries to break the product down into its prime factors – even using the fastest supercomputer – no known algorithm exists that can accomplish that before the sun will explode. </p>
<h2>The primal quest</h2>
<p>Large prime numbers are used prominently in other cryptosystems too. The faster computers get, the larger the numbers they can crack. For modern applications, prime numbers measuring hundreds of digits suffice. These numbers are minuscule in comparison to the giant recently discovered. In fact, the new prime is so large that – at present – no conceivable technological advancement in computing speed could lead to a need to use it for cryptographic safety. It is even likely that the risks posed by the looming quantum computers wouldn’t need such monster numbers to be made safe. </p>
<p>It is neither safer cryptosystems nor improving computers that drove the latest Mersenne discovery, however. It is mathematicians’ need to uncover the jewels inside the chest labelled “prime numbers” that fuels the ongoing quest. This is a primal desire that starts with counting one, two, three, and drives us to the frontiers of research. The fact that online commerce has been revolutionised is almost an accident.</p>
<p>The celebrated British mathematician <a href="https://theconversation.com/the-man-who-taught-infinity-how-gh-hardy-tamed-srinivasa-ramanujans-genius-57585">Godfrey Harold Hardy</a> said: “Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics”. Whether or not huge prime numbers, such as the 50th known Mersenne prime with its millions of digits, will ever be found useful is, at least to Hardy, an irrelevant question. The merit of knowing these numbers lies in quenching the human race’s intellectual thirst that started with Euclid’s proof of the infinitude of primes and still goes on today.</p><img src="https://counter.theconversation.com/content/89878/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ittay Weiss does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Prime numbers are a mathematical mystery.Ittay Weiss, Teaching Fellow, Department of Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/641412016-09-19T15:14:07Z2016-09-19T15:14:07ZWhat's the point of maths research? It's the abstract nonsense behind tomorrow's breakthroughs<figure><img src="https://images.theconversation.com/files/138241/original/image-20160919-11113-1j3jacl.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Whenever I tell people I’m a mathematical researcher, I’m usually met with some form of bewilderment. Occasionally that’s followed by the immediate end of the conversation. If there is a follow-up question, it’s usually not about the type of research I’m doing or how it’s funded but whether there’s anything left to discover in maths at all.</p>
<p>True, maths rarely makes the headlines and so most people probably don’t think of it as carrying out cutting-edge research. But neither does, say, geology and people don’t assume there’s nothing left to discover in that field. The difference is that everyone is familiar with maths from their schooldays in a way that contrasts vastly from the work of actual mathematicians. In school we learn formulas that are then used to calculate answers to specific problems. The right method correctly calculated will never fail.</p>
<p>Maths research, on the other hand, looks at the myriad problems for which we don’t have such a method. It’s about finding the tools and systems that other subject areas find so useful in formulating their own work. And sometimes it stumbles across <a href="http://mathworld.wolfram.com/RamseyNumber.html">facts about numbers</a> that we have no conceivable use for at the moment but that one day could become vital to the world.</p>
<p>Any mathematical method used at school (or work or anywhere) was figured out at some point by a mathematician. Another mathematician may have proven that it always works. And another may have worked out how to use the method in the real world. Someone else might then have shown that it’s not a very efficient way to solve larger problems and developed a different approach instead.</p>
<p>The method may also have relied on several properties of the underlying number system discovered over a long stretch of time. Others before them will have accomplished the important but unglamorous task of precisely defining that number system, perhaps <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/History_overview.html">a very long time ago</a>.</p>
<h2>Prime purpose</h2>
<p>Research mathematicians essentially still discover similar types of results today. We have simply moved on to different questions that have become important, to new methods for existing questions, to different systems that draw our attention, and to more advanced questions about things that have already been researched.</p>
<p>Here is an example of such a recent result. It deals with the distribution of prime numbers, like 7, 11, 23, or 37, which you cannot divide by another natural number other than 1 or themselves. We’ve found prime numbers as large as <a href="http://www.bbc.co.uk/news/technology-35361090">22m digits</a> long, and researchers are <a href="http://www.mersenne.org">still looking</a>.</p>
<p>If you look at a table of numbers, the prime numbers seem to be almost randomly mixed into the non-prime ones. For a long time, we have been able to describe the <a href="https://primes.utm.edu/howmany.html#better">typical characteristics</a> of prime numbers. As it turns out, prime numbers slowly but steadily appear less frequently – they “thin out” – among the larger numbers. What’s more, we can quantify this process in surprisingly precise terms. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/138242/original/image-20160919-11100-o96axa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/138242/original/image-20160919-11100-o96axa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/138242/original/image-20160919-11100-o96axa.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/138242/original/image-20160919-11100-o96axa.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/138242/original/image-20160919-11100-o96axa.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/138242/original/image-20160919-11100-o96axa.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/138242/original/image-20160919-11100-o96axa.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">‘Useless’ number theory turned out to have a very important purpose for modern society.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>With the average primes being further and further apart as we look at ever larger numbers, a typical question for maths researchers to ask is whether this process of thinning out also carries over to the smallest gaps between primes. In other words, will all large primes come at increasing distances from each other, or will we always find primes that are close to each other?</p>
<p><a href="http://annals.math.princeton.edu/2014/179-3/p07">A breakthrough result</a> in 2014 showed that no matter how high we go among the numbers, we will always find two primes that are closer to each other than some constant number. That number was initially a whopping 70m. This might not seem very close but the fact that we could identify a finite number was an important breakthrough. Other mathematicians then set out to reduce this value, and the best I am currently aware of is a <a href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes">much more manageable 246</a>.</p>
<h2>Real applications – eventually</h2>
<p>You might wonder how solving such abstract problems helps anyone outside of mathematics. First, there is a trickle-down effect. A fundamental result is useful in obtaining other pure mathematical results, which in turn are used to develop applied mathematics, which are then used by non-mathematicians. Second – and more importantly – mathematical theory is often ahead of its time, and the abstract nonsense of yesterday underpins the applied mathematics of today.</p>
<p>For example, number theory is the area that examines, among other items, questions like our prime number example. For many years this was considered the ultimate pure mathematics topic and completely unusable for any purpose other than satisfying human curiosity. The eminent early 20th-century number theorist and pacifist, G.H Hardy, was <a href="http://www.cambridge.org/gb/academic/subjects/mathematics/recreational-mathematics/mathematicians-apology-2?format=PB&isbn=9781107604636">very proud to say</a>: “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” In other words, he was glad his work could not be used for military purposes.</p>
<p>Nowadays, the number theory results that seemed so useless less than a century ago <a href="http://dl.acm.org/citation.cfm?id=359342">are at the heart</a> of the encryption algorithms that let us securely order a product or check our bank accounts online. In a way that would have horrified Hardy, British intelligence services had actually already <a href="http://simonsingh.net/media/articles/maths-and-science/unsung-heroes-of-cryptography/">discovered the same method</a> in secret ahead of their civil colleagues. </p>
<p>When the next technological or scientific breakthrough requires a new type of mathematical model, it is likely that the subject already has the underlying theory in hand, waiting to be adapted to a new setting.</p>
<p>Underlying all of this is one of the fundamental truths about mathematical research. The applications of mathematics might change with scientific progress, making some mathematical topics more useful at times than others. But because mathematical results are based on logical deductions alone, they actually never become wrong, never get obsolete, and never truly get old. They are just waiting for the right application to arrive.</p><img src="https://counter.theconversation.com/content/64141/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Wolfram Bentz does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>We don't know what knowledge we'll need in the future, and that's where maths research comes in.Wolfram Bentz, Director of Research for Mathematics, University of HullLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/534592016-01-20T16:40:36Z2016-01-20T16:40:36ZThe 22 million digit number ... and the amazing maths behind primes<figure><img src="https://images.theconversation.com/files/108760/original/image-20160120-26085-gwefom.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Primes: here be magic.</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>It is a quite extraordinary figure. Dr Curtis Cooper from the University of Central Missouri has found the <a href="http://www.bbc.co.uk/news/technology-35361090">largest-known prime number</a> – written (2<sup>74207281)-1.</sup> It is around 22m digits long and, if printed in full, would take you days to read. Its discovery comes thanks to a collaborative project of volunteers who use freely available software called <a href="http://www.mersenne.org/various/history.php">GIMPS</a> (Great Internet Mersenne Prime Search) to search for primes. </p>
<p>A number which can only be divided by itself and 1 without a remainder is called a <a href="https://www.mathsisfun.com/prime_numbers.html">prime number</a>. Here is a list of the primes less than 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. </p>
<h2>Unlucky 13</h2>
<p>Numbers appear everywhere in our lives – and good and bad superstitions have developed out of them. Remarkably, most of these superstitious numbers are prime. The superstition that 13 is unlucky results in some hotels and office buildings <a href="http://www.huffingtonpost.com/orbitzcom/unlucky-13-the-truth-behind-hotels-missing-13th-floors_b_6848290.html">not having rooms or floors labelled 13</a>. And we all fear <a href="http://www.theguardian.com/science/brain-flapping/2015/feb/13/friday-13th-unlucky-why-science-psychology">Friday 13th</a>, especially sufferers of <a href="http://skepdic.com/paraskevidekatriaphobia.html">paraskevidekatriaphobia</a>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/108752/original/image-20160120-26101-1qmlc2r.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/108752/original/image-20160120-26101-1qmlc2r.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=425&fit=crop&dpr=1 600w, https://images.theconversation.com/files/108752/original/image-20160120-26101-1qmlc2r.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=425&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/108752/original/image-20160120-26101-1qmlc2r.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=425&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/108752/original/image-20160120-26101-1qmlc2r.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=534&fit=crop&dpr=1 754w, https://images.theconversation.com/files/108752/original/image-20160120-26101-1qmlc2r.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=534&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/108752/original/image-20160120-26101-1qmlc2r.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=534&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Unlucky for some.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/sidelong/41800177/in/photolist-4GeJH-nFeAP8-GrrR8-nF7diR-8rZvvE-nF2tKJ-nF6ryd-5ZyLyi-o3gNMV-nF4p9r-nUYfkf-nF4Eif-nXs5E5-8soYhH-nYCiX3-o1EPPR-5ZyKvS-8scNo8-8sgbQ9-9HJ98N-8scBtV-dRAyj7-dRAHeE-dRAvp9-dRABCE-9HhTi1-dRuZvn-dRuFGP-dRuWbx-dRuE8e-dRuCJH-dRAnAA-dRuJJR-dRuPWF-dRAGiy-dRAxS9-dRuXb2-dRAckE-dRAzV3-dRAvQC-dRAbdb-dRuYzM-dRuSQn-dRAk8f-dRAeUd-dRAmsU-dRAjEq-dRAgRy-dRAsZY-dRuRBx">Dave Bleasdale/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>The most popular explanation for 13 being unlucky is that at the last supper there was Jesus and the Twelve Apostles, with the 13th guest being <a href="http://www.britannica.com/biography/Judas-Iscariot">Judas Iscariot</a> who went on to betray Jesus.</p>
<p>The number 3 also has religious significance and references to it can be found not only in the Holy Trinity of Father, Son and Holy Ghost, but also the Three Wise Men and in the architectural structures of churches. There is also a superstitious fear of <a href="http://www.dailymail.co.uk/femail/article-2230328/Britons-superstitions-Walking-ladders-breaking-mirrors-opening-umbrellas-indoors.html">walking under a ladder</a>, which seems to have its origins in the number 3. Propped against a wall, a ladder forms the longest side of a triangle, with the ground and the wall forming the other two sides. A person passing under the ladder is symbolically breaking the Trinity and thus brings bad luck on themselves.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/108753/original/image-20160120-26125-krgrzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/108753/original/image-20160120-26125-krgrzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=786&fit=crop&dpr=1 600w, https://images.theconversation.com/files/108753/original/image-20160120-26125-krgrzc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=786&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/108753/original/image-20160120-26125-krgrzc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=786&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/108753/original/image-20160120-26125-krgrzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=987&fit=crop&dpr=1 754w, https://images.theconversation.com/files/108753/original/image-20160120-26125-krgrzc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=987&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/108753/original/image-20160120-26125-krgrzc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=987&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Don’t do it!</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/dl2_lim.mhtml?src=auobXiF4tpx28llbbAg2Gg-1-1&clicksrc=download_btn_inline&id=92578438&size=huge_jpg&submit_jpg=">Shutterstock</a></span>
</figcaption>
</figure>
<h2>Big rewards</h2>
<p>Mathematicians have been searching for patterns in prime numbers for more than 3,000 years and have made only a small amount of progress, believing that there are still many patterns to find. This recent discovery continues that pursuit of understanding.</p>
<p>But why? Well, you could be doing it for money. <a href="http://www.claymath.org/">The Clay Mathematics Institute</a> is offering a million dollars to anyone who can solve the “<a href="http://www.claymath.org/millennium-problems/riemann-hypothesis">Riemann problem</a>”. This is a complex mathematics puzzle that emerged from the attempts by mathematicians to understand the intricacies of prime numbers. And so finding larger primes, some believe, may help in this quest. </p>
<p>Or maybe you are just looking for “the truth”, something mathematicians have been doing for a very long time. <a href="http://www.britannica.com/biography/Eratosthenes-of-Cyrene">Eratosthenes</a> was a Greek mathematician who was working at the library in Alexandra around 200BC when he discovered the first method of listing primes. </p>
<p>He was very keen on all types of learning (his nickname was Philogus, or “the one who loves learning”). He called his method “the sieve”, as primes just fall out when you apply it – and it offers a flavour of prime searching. </p>
<p>First – and this begins to get technical – note that if a number is a composite, such as n=ab, then a and b cannot both exceed √n. For example, with the composite “21” – 21=3x7 – only 7 is bigger than √21 = 4.58. Therefore, he determined that any composite integer n is divisible by a prime p that does not exceed √n. </p>
<p>It follows from this that to test for primes it is only necessary to divide a number by numbers less than or equal to its square root. To find primes from 2 to 30, then, we need only use the fact that √30 is less than 7, and work with the primes 2, 3 and 5.</p>
<p>So if you write out the list of numbers from 2 to 30 on a piece of paper, we can “sieve” out any numbers that are divisible by 2, 3 and 5 to leave us with the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. </p>
<h2>Mysterious numbers</h2>
<p>Primes are strange and curious numbers. There are, for example, no primes between 370,261 and 370,373, or between 20,831,323 and 20,831,533. And the primes 13,331, 15,551, 16,661, 19,991 and 72,227 and 1,777,771 are all examples of palindromic numbers. These are numbers that remain the same when the digits are reversed. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/108769/original/image-20160120-26085-iyeyvs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/108769/original/image-20160120-26085-iyeyvs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=398&fit=crop&dpr=1 600w, https://images.theconversation.com/files/108769/original/image-20160120-26085-iyeyvs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=398&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/108769/original/image-20160120-26085-iyeyvs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=398&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/108769/original/image-20160120-26085-iyeyvs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=501&fit=crop&dpr=1 754w, https://images.theconversation.com/files/108769/original/image-20160120-26085-iyeyvs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=501&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/108769/original/image-20160120-26085-iyeyvs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=501&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Magic number 7.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/morberg/3300564928/in/photolist-62EfXS-jXgNBC-cXoRCW-dNrHAz-dNmM2F-aMW1ep-fb6gfe-qGCBBd-fiygo2-5ZkmPQ-7QaXLd-at5uUd-pw8esG-vQHWPT-aYWr8F-qrpceD-89fiqp-8RbcFy-vXMMib-kKzgp-yp2frC-8bWjbk-7aYKNF-6nSHep-dg5M1c-2MiqBy-omHQyy-xeg6Ey-6nWSVw-KUKpL-8s2qzq-jHaBen-rLg5D1-2vp7v1-9u7Mpb-c3n42-5TTzwX-4VqfDK-7TgVUC-28UFTB-rZSEE4-4JALqn-nATji-MBXNe-7yf88a-fUrMM-dcSHmT-e5Qj5B-dL6xpo-gqo7r2">Niklas Morberg/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>In 1956, psychologist George A Miller published a paper in The Psychological Review called <a href="https://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_or_Minus_Two">The Magical Number Seven, Plus or Minus Two</a>. In the paper, he talks about the prime number 7 “following him around”. Religion, for example, is filled with sevens, from the Seven Deadly Sins to the Seven Sacraments. And salesman believe in the “<a href="http://www.effectivebusinessideas.com/the-rule-of-7/">rule of seven</a>”, which suggests people need to hear a marketing message seven times before they take action. Miller, however, claims that this is more than just coincidence. </p>
<p>Our immediate memory has been shown to perform well when remembering up to, but no more than, seven things. We can distinguish and make a judgement about seven different categories. Our span of attention will also remember around seven different objects at a glance. Miller also looked into other areas of how we record and store information and found to his surprise that seven appeared over and over. In conclusion, Miller makes no claim that this is something deep and profound, but says maybe, just maybe, seven could be more special than we had imagined and needs a closer look. </p>
<p>Prime numbers are interesting, don’t you think?</p><img src="https://counter.theconversation.com/content/53459/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Steve Humble does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The largest known prime number has been discovered. But what does it all mean?Steve Humble, Researcher in International Development and Education, and Head of PGCE Maths teacher training for Primary and Secondary Education, Newcastle UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/403272015-06-07T23:16:35Z2015-06-07T23:16:35ZA little number theory makes the times table a thing of beauty<figure><img src="https://images.theconversation.com/files/83790/original/image-20150603-10701-1j85hpt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Doing the 9 times table.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/jimmiehomeschoolmom/4007733107">Flickr/Jimmie</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span></figcaption></figure><p>Most people will probably remember the times tables from primary school quizzes. There might be patterns in some of them (the simple doubling of the 2 times table) but others you just learnt by rote. And it was never quite clear just why it was necessary to know what 7 x 9 is off the top of your head.</p>
<p>Well, have no fear, there will be no number quizzes here.</p>
<p>Instead, I want to show you a way to build numbers that gives them some structure, and how multiplication uses that structure.</p>
<h2>Understanding multiplication</h2>
<p>Multiplication simply gives you the area of a rectangle, if you know the lengths of the sides. Pick any square in the grid, (for example, let’s pick the 7th entry in the 5th row) and colour a rectangle from that square to the top left corner.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A rectangle of size 5 × 7 in the multiplication table.</span>
</figcaption>
</figure>
<p>This rectangle has length 7 and height 5, and the area (the number of green squares) is found in the blue circle in the bottom right corner! This is true no matter which pair of numbers in the grid you pick.</p>
<p>Now let’s take this rectangle and flip it around the main diagonal (the red dotted line).</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The same rectangle, flipped.</span>
</figcaption>
</figure>
<p>The length and height of the rectangle have swapped, but the area hasn’t changed. So from this we can see that 5 × 7 is the same as 7 × 5. This holds true for any pair of numbers — in mathematics we say that multiplication is commutative.</p>
<p>But this fact means that there is a symmetry in the multiplication table. The numbers above the diagonal line are like a mirror image of the numbers below the line.</p>
<p>So if your aim is to memorise the table, you really only need to memorise about half of it.</p>
<h2>The building blocks of numbers</h2>
<p>To go further with multiplication we first need to do some dividing. Remember that dividing a number just means breaking it into pieces of equal size.</p>
<blockquote>
<p>12 ÷ 3 = 4</p>
</blockquote>
<p>This means 12 can be broken into 3 pieces, each of size 4.</p>
<p>Since 3 and 4 are both whole numbers, they are called factors of 12, and 12 is said to be divisible by 3 and by 4. If a number is only divisible by itself and 1, it is called a prime number.</p>
<p>But there’s more than one way to write 12 as a product of two numbers:</p>
<blockquote>
<p>12 × 1</p>
<p>6 × 2</p>
<p>4 × 3</p>
<p>3 × 4</p>
<p>2 × 6</p>
<p>1 × 12</p>
</blockquote>
<p>In fact, we can see this if we look at the multiplication table.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The occurrences of 12 in the multiplication table.</span>
</figcaption>
</figure>
<p>The number of coloured squares in this picture tells you there are six ways you can make a rectangle of area 12 with whole number side lengths. So it’s also the number of ways you can write 12 as a product of two numbers.</p>
<p>Incidentally, you might have noticed that the coloured squares seem to form a smooth curve — they do! The curve joining the squares is known as a hyperbola, given by the equation a × b = 12, where ‘a’ and ‘b’ are not necessarily whole numbers.</p>
<p>Let’s look again at the list of products above that are equal to 12. Every number listed there is a factor of 12. What if we look at factors of factors? Any factor that is not prime (except for 1) can be split into further factors, for example</p>
<blockquote>
<p>12 = 6 × 2 = (2 × 3) × 2</p>
<p>12 = 4 × 3 = (2 × 2) × 3</p>
</blockquote>
<p>No matter how we do it, when we split the factors until we’re left only with primes, we always end up with two 2’s and one 3.</p>
<p>This product</p>
<blockquote>
<p>2 × 2 × 3</p>
</blockquote>
<p>is called the prime decomposition of 12 and is unique to that number. There is only one way to write a number as a product of primes, and each product of primes gives a different number. In mathematics this is known as the <a href="https://www.mathsisfun.com/numbers/fundamental-theorem-arithmetic.html">Fundamental Theorem of Arithmetic</a>.</p>
<p>The prime decomposition tells us important things about a number, in a very condensed way.</p>
<p>For example, from the prime decomposition 12 = 2 × 2 × 3, we can see immediately that 12 is divisible by 2 and 3, and not by any other prime (such as 5 or 7). We can also see that it’s divisible by the product of any choice of two 2’s and one 3 that you want to pick.</p>
<p>Furthermore, any multiple of 12 will also be divisible by the same numbers. Consider 11 x 12 = 132. This result is also divisible by 1, 2, 3, 4, 6 and 12, just like 12. Multiplying each of these with the factor of 11, we find that 132 is also divisible by 11, 22, 33, 44, 66 and 132.</p>
<p>It’s also easy to see if a number is the square of another number: In that case there must be an even number of each prime factor. For example, 36 = 2 × 2 × 3 × 3, so it’s the square of 2 × 3 = 6.</p>
<p>The prime decomposition can also make multiplication easier. If you don’t know the answer to 11 × 12, then knowing the prime decomposition of 12 means you can work through the multiplication step by step.</p>
<blockquote>
<p>11 x 12</p>
<p>= 11 x 2 × 2 × 3</p>
<p>= ((11 x 2) × 2) × 3</p>
<p>= (22 × 2) × 3</p>
<p>= 44 × 3</p>
<p>= 132</p>
</blockquote>
<p>If the primes of the decomposition are small enough (say 2, 3 or 5), multiplication is nice and easy, if a bit paper-consuming. Thus multiplying by 4 (= 2 x 2), 6 (= 2 x 3), 8 (= 2 x 2 x 2), or 9 (= 3 x 3) doesn’t need to be a daunting task!</p>
<p>For example, if you can’t remember the 9 times table, it doesn’t matter as long as you can multiply by 3 twice. (However this method doesn’t help with multiplying by larger primes, here new methods are required – if you haven’t seen the trick for the 11 times tables <a href="https://www.youtube.com/watch?v=rRzOWmyulS4">watch this video</a>).</p>
<p>So the ability to break numbers into their prime factors can make complicated multiplications much simpler, and it’s even more useful for bigger numbers.</p>
<p>For example, the prime decomposition of 756 is 2 x 2 x 3 x 3 x 3 x 7, so multiplying by 756 simply means multiplying by each of these relatively small primes. (Of course, finding the prime decomposition of a large number is usually very difficult, so it’s only useful if you already know what the decomposition is.)</p>
<p>But more than this, prime decompositions give fundamental information about numbers. This information is widely useful in mathematics and other fields such as cryptography and internet security. It also leads to some surprising patterns – to see this, try colouring all multiples of 12 in the times table and see what happens. I’ll leave that for homework.</p><img src="https://counter.theconversation.com/content/40327/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Anita Ponsaing receives funding from the ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS).</span></em></p>If you know a bit about how numbers are made then you don't need to work out all 144 calculations in a 12 by 12 times table.Anita Ponsaing, Research Associate in Mathematics, University of MelbourneLicensed as Creative Commons – attribution, no derivatives.