Author: IMC

Is there an error in there somewhere?

It’s the stuff of Hollywood. Somebody somewhere is surely selling the movie rights to what’s become the biggest spat in maths: a misunderstood genius, a 500-page proof almost nobody can understand and a supporting cast squabbling over what it all means. At stake: nothing less than the future of pure mathematics.

In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a proof of a long-standing problem called the ABC conjecture. Six years later the jury is still out on whether it’s correct. But in a new twist, Peter Scholze at the University of Bonn – who was awarded the Fields Medal, the highest honour in maths, in August – and Jakob Stix at Goethe University Frankfurt – who is an expert in the type of maths used by Mochizuki – claim to have found an error at the heart of Mochizuki’s proof.

Roll credits? Not so fast. The pairs’ reputation means that their claim is a serious blow for Mochizuki. And a handful of other mathematicians claim to have lost the thread of the proof at the same point Scholze and Stix say there is an error. But there is still room for dispute.

a + b = c?

The ABC conjecture was first proposed in the 1980s and concerns a fundamental property of numbers, based around the simple equation a + b = c. For a long time, mathematicians believed that the conjecture was true but nobody had ever been able to prove it.

To tackle the problem, Mochizuki had to invent a fiendish type of maths called Inter-universal Teichmüller (IUT) theory. In an effort to understand IUT better, Scholze and Stix spent a week with Mochizuki in Tokyo in March. By the end of the week, they claim to have found an error.

The alleged flaw comes in Conjecture 3.12, which many see as the crux of the proof. This section involves measuring an equivalence between different mathematical objects. In effect, Scholze and Stix claim that Mochizuki changes the length of the measuring stick in the middle of the process.

No proof

“We came to the conclusion that there is no proof,” they write in their report, which was posted online on 20 September.

But Ivan Fesenko at the University of Nottingham, UK, who says he is one of only 15 people around the world who actually understand Mochizuki’s theory, thinks Scholze and Stix are jumping the gun. “They spent much less time than all of us who have been studying this for many years,” says Fesenko.

Mochizuki has tried to help others understand his work, taking part in seminars and answering questions. Mochizuki was even the one who posted Scholze and Stix’s critical report. “We have this paradoxical situation in which the victim has published the report of the villain,” says Fesenko with a laugh. “This is an unprecedented event in mathematics.”

So is the proof wrong or just badly explained? Fesenko thinks that the six-year dispute exposes something rotten at the heart of pure mathematics. These days mathematicians work in very narrow niches, he says. “People just do not understand what the mathematician in the next office to you is doing.”

This means that mathematicians will increasingly have to accept others’ proofs without actually understanding them – something Fesenko describes as a fundamental problem for the future development of mathematics.

This suggests the story of Mochizuki’s proof may forever lack a satisfactory ending – becoming a war between mathematicians that is doomed to spiral into infinity. “My honest answer is that we will never have consensus about it,” says Fesenko.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Douglas Heaven*

 

For millennia mathematicians have struggled to unify arithmetic and geometry. Now one young genius could have brought them in sight of the ultimate prize.

IF JOEY was Chloe’s age when he was twice as old as Zoe was, how many times older will Zoe be when Chloe is twice as old as Joey is now?

Or try this one for size. Two farmers inherit a square field containing a crop planted in a circle. Without knowing the exact size of the field or crop, or the crop’s position within the field, how can they draw a single line to divide both the crop and field equally?

You’ve either fallen into a cold sweat or you’re sharpening your pencil (if you can’t wait for the answer, you can check the bottom of this page). Either way, although both problems count as “maths” – or “math” if you insist – they are clearly very different. One is arithmetic, which deals with the properties of whole numbers: 1, 2, 3 and so on as far as you can count. It cares about how many separate things there are, but not what they look like or how they behave. The other is geometry, a discipline built on ideas of continuity: of lines, shapes and other objects that can be measured, and the spatial relationships between them.

Mathematicians have long sought to build bridges between these two ancient subjects, and construct something like a “grand unified theory” of their discipline. Just recently, one brilliant young researcher might have brought them decisively closer. His radical new geometrical insights might not only unite mathematics, but also help solve one of the deepest number problems of them all: the riddle of the primes. With the biggest prizes in mathematics, the Fields medals, to be awarded this August, he is beginning to look like a shoo-in.

The ancient Greek philosopher and mathematician Aristotle once wrote, “We cannot… prove geometrical truths by arithmetic.” He left little doubt he believed geometry couldn’t help with numbers, either. It was hardly a controversial thought for the time. The geometrical proofs of Aristotle’s near-contemporary Euclid, often called the father of geometry, relied not on numbers, but logical axioms extended into proofs by drawing lines and shapes. Numbers existed on an entirely different, more abstract plane, inaccessible to geometers’ tools.

And so it largely remained until, in the 1600s, the Frenchman René Descartes used the techniques of algebra – of equation-solving and the manipulation of abstract symbols – to put Euclid’s geometry on a completely new footing. By introducing the notion that geometrical points, lines and shapes could all be described by numerical coordinates on an underlying grid, he allowed geometers to make use of arithmetic’s toolkit, and solve problems numerically.

This was a moonshot that let us, eventually, do things like send rockets into space or pinpoint positions to needle-sharp accuracy on Earth. But to a pure mathematician it is only a halfway house. A circle, for instance, can be perfectly encapsulated by an algebraic equation. But a circle drawn on graph paper, produced by plotting out the equation’s solutions, would only ever capture a fragment of that truth. Change the system of numbers you use, for example – as a pure mathematician might do – and the equation remains valid, while the drawing may no longer be helpful.

Wind forward to 1940 and another Frenchman was deeply exercised by the divide between geometry and numbers. André Weil was being held as a conscientious objector in a prison just outside Rouen, having refused to enlist in the months preceding the German occupation of France – a lucky break, as it turned out. In a letter to his wife, he wrote: “If it’s only in prison that I work so well, will I have to arrange to spend two or three months locked up every year?”

Weil hoped to find a Rosetta stone between algebra and geometry, a reference work that would allow truths in one field to be translated into the other. While behind bars, he found a fragment.

It had to do with the Riemann hypothesis, a notorious problem concerning how those most fascinating numbers, the primes, are distributed (see below). There had already been hints that the hypothesis might have geometrical parallels. Back in the 1930s, a variant had been proved for objects known as elliptic curves. Instead of trying to work out how prime numbers are distributed, says mathematician Ana Caraiani at Imperial College London, “you can relate it to asking how many points a curve has”.

Weil proved that this Riemann-hypothesis equivalent applied for a range of more complicated curves too. The wall that had stood between the two disciplines since Ancient Greek times finally seemed to be crumbling. “Weil’s proof marks the beginning of the science with the most un-Aristotelian name of arithmetic geometry,” says Michael Harris of Columbia University in New York.

The Riemann Hypothesis: The million-dollar question

The prime numbers are the atoms of the number system, integers indivisible into smaller whole numbers other than one. There are an infinite number of them and there is no discernible pattern to their appearance along the number line. But their frequency can be measured – and the Riemann hypothesis, formulated by Bernhard Riemann in 1859, predicts that this frequency follows a simple rule set out by a mathematical expression now known as the Riemann zeta function.

Since then, the validity of Riemann’s hypothesis has been demonstrated for the first 10 trillion primes, but an absolute proof has yet to emerge. As a mark of the problem’s importance, it was included in the list of seven Millennium Problems set by the Clay Mathematics Institute in New Hampshire in 2000. Any mathematician who can tame it stands to win $1 million.

In the post-war years, in the more comfortable setting of the University of Chicago, Weil tried to apply his insight to the broader riddle of the primes, without success. The torch was taken up by Alexander Grothendieck, a mathematician ranked as one of the greatest of the 20th century. In the 1960s, he redefined arithmetic geometry.

Among other innovations, Grothendieck gave the set of whole numbers what he called a “spectrum”, for short Spec(Z). The points of this undrawable geometrical entity were intimately connected to the prime numbers. If you could ever work out its overall shape, you might gain insights into the prime numbers’ distribution. You would have built a bridge between arithmetic and geometry that ran straight through the Riemann hypothesis.

The shape Grothendieck was seeking for Spec(Z) was entirely different from any geometrical form we might be familiar with: Euclid’s circles and triangles, or Descartes’s parabolas and ellipses drawn on graph paper. In a Euclidean or Cartesian plane, a point is just a dot on a flat surface, says Harris, “but a Grothendieck point is more like a way of thinking about the plane”. It encompasses all the potential uses to which a plane could be put, such as the possibility of drawing a triangle or an ellipse on its surface, or even wrapping it map-like around a sphere.

If that leaves you lost, you are in good company. Even Grothendieck didn’t manage to work out the geometry of Spec(Z), let alone solve the Riemann hypothesis. That’s where Peter Scholze enters the story.

“Even the majority of mathematicians find most of the work unintelligible”

Born in Dresden in what was then East Germany in 1987, Scholze is currently, at the age of 30, a professor at the University of Bonn. He laid the first bricks for his bridge linking arithmetic and geometry in his PhD dissertation, published in 2012 when he was 24. In it, he introduced an extension of Grothendieck-style geometry, which he termed perfectoid geometry. His construction is built on a system of numbers known as the p-adics that are intimately connected with the prime numbers (see “The p-adics: A different way of doing numbers”). The key point is that in Scholze’s perfectoid geometry, a prime number, represented by its associated p-adics, can be made to behave like a variable in an equation, allowing geometrical methods to be applied in an arithmetical setting.

It’s not easy to explain much more. Scholze’s innovation represents “one of the most difficult notions ever introduced in arithmetic geometry, which has a long tradition of difficult notions”, says Harris. Even the majority of working mathematicians find most of it unintelligible, he adds.

Be that as it may, in the past few years, Scholze and a few initiates have used the approach to solve or clarify many problems in arithmetic geometry, to great acclaim. “He’s really unique as a mathematician,” says Caraiani, who has been collaborating with him. “It’s very exciting to be a mathematician working in the same field.”

This August, the world’s mathematicians are set to gather in Rio de Janeiro, Brazil, for their latest international congress, a jamboree held every four years. A centrepiece of the event is the awarding of the Fields medals. Up to four of these awards are given each time to mathematicians under the age of 40, and this time round there is one name everyone expects to be on the list. “I suspect the only way he can escape getting a Fields medal this year is if the committee decides he’s young enough to wait another four years,” says Marcus du Sautoy at the University of Oxford.

 

Peter Scholze, 30, looks like a shoo-in for mathematics’s highest accolade this summer

With so many grand vistas opening up, the question of Spec(Z) and the Riemann hypothesis almost becomes a sideshow. But Scholze’s new methods have allowed him to study the geometry, in the sense Grothendieck pioneered, that you would see if you examined the curve Spec(Z) under a microscope around the point corresponding to a prime number p. That is still a long way from understanding the curve as a whole, or proving the Riemann hypothesis, but his work has given mathematicians hope that this distant goal might yet be reached. “Even this is a huge breakthrough,” says Caraiani.

Scholze’s perfectoid spaces have enabled bridges to be built in entirely different directions, too. A half-century ago, in 1967, the then 30-year-old Princeton mathematician Robert Langlands wrote a tentative letter to Weil outlining a grand new idea. “If you are willing to read it as pure speculation I would appreciate that,” he wrote. “If not – I am sure you have a waste basket handy.”

In his letter, Langlands suggested that two entirely distinct branches of mathematics, number theory and harmonic analysis, might be related. It contained the seeds of what became known as the Langlands program, a vastly influential series of conjectures some mathematicians have taken to calling a grand unified theory capable of linking the three core mathematical disciplines: arithmetic, geometry and analysis, a broad field that we encounter in school in the form of calculus. Hundreds of mathematicians around the world, including Scholze, are committed to its completion.

The full slate of Langlands conjectures is no more likely than the original Riemann hypothesis to be proved soon. But spectacular discoveries could lie in store: Fermat’s last theorem, which took 350 years to prove before the British mathematician Andrew Wiles finally did so in 1994, represents just one particular consequence of its conjectures. Recently, the French mathematician Laurent Fargues proposed a way to build on Scholze’s work to understand aspects of the Langlands program concerned with p-adics. It is rumoured that a partial solution could appear in time for the Rio meeting.

In March, Langlands won the other great mathematical award, the Abel prize, for his lifetime’s work. “It took a long time for the importance of Langlands’s ideas to be recognised,” says Caraiani, “and they were overdue for a major award.” Scholze seems unlikely to have to wait so long.

The p-adics: A different way of doing numbers

Key to the latest work in unifying arithmetic and geometry are p-adic numbers.

These are an alternative way of representing numbers in terms of any given prime number p. To make a p-adic number from any positive integer, for example, you write that number in base p, and reverse it. So to write 20 in 2-adic form, say, you take its binary, or base-2, representation – 10100 – and write it backwards, 00101. Similarly 20’s 3-adic equivalent is 202, and as a 4-adic it is written 011.

The rules for manipulating p-adics are a little different, too. Most notably, numbers become closer as their difference grows more divisible by whatever p is. In the 5-adic numbers, for example, the equivalents of 11 and 36 are very close because their difference is divisible by 5, whereas the equivalents of 10 and 11 are further apart.

For decades after their invention in the 1890s, the p-adics were just a pretty mathematical toy: fun to play with, but of no practical use. But in 1920, the German mathematician Helmut Hasse came across the concept in a pamphlet in a second-hand bookshop, and became fascinated. He realised that the p-adics provided a way of harnessing the unfactorisability of the primes – the fact they can’t be divided by other numbers – that turned into a shortcut to solving complicated proofs.

Since then, p-adics have played a pivotal part in the branch of maths called number theory. When Andrew Wiles proved Fermat’s infamous last theorem (that the equation xⁿ + yⁿ = zⁿ has no solutions when x, y and z are positive integers and n is an integer greater than 2) in the early 1990s, practically every step in the proof involved p-adic numbers.

• Answers: Zoe will be three times as old as she is now. The farmers should draw a line across the field that connects the centre points of the field and the crop.

This article appeared in print under the headline “The shape of numbers”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Gilead Amit*

Have you ever come across the words “commutative diagram” before? Perhaps you’ve read or heard someone utter a sentence that went something like

“For every [bla bla] there existsa [yadda yadda] such that the following diagram commutes.”

and perhaps it left you wondering what it all meant. I was certainly mystified when I first came across a commutative diagram. And it didn’t take long to realize that the phrase “the following diagram commutes” was (and is) quite ubiquitous. It appears in theorems, propositions, lemmas and corollaries almost everywhere!

So what’s the big deal with diagrams? And what does commute mean anyway?? It turns out the answer is quite simple.

Do you know what a composition of two functions is?

Then you know what a commutative diagram is!

A commutative diagram is simply the picture behind function composition.

Truly, it is that simple. To see this, suppose AA and BB are sets and ff is a function from AA to BB. Since ff maps (i.e. assigns) elements in AA to elements in BB, it is often helpful to denote that process by an arrow.

And there you go. That’s an example of a diagram. But suppose we have another function gg from sets BB to CC, and suppose ff and gg are composable. Let’s denote their composition by h=g∘fh=g∘f. Then both gg and hh can be depicted as arrows, too.

But what is the arrow A→CA→C really? I mean, really? Really it’s just the arrows ff and gg lined up side-by-side.

But maybe we think that drawing hh’s arrow curved upwards like that takes up too much space, so let’s bend the diagram a bit and redraw it like this:

This little triangle is the paradigm example of a commutative diagram. It’s a diagram because it’s a schematic picture of arrows that represent functions. And it commutes because the diagonal function IS EQUAL TO the composition of the vertical and horizontal functions, i.e. h(a)=g(f(a))h(a)=g(f(a)) for every a∈Aa∈A. So a diagram “commutes” if all paths that share a common starting and ending point are the same. In other words, your diagram commutes if it doesn’t matter how you commute from one location to another in the diagram.

But be careful.

Not every diagram is a commutative diagram.

The picture on the right is a bona fide diagram of real-valued functions, but it is defintitely not commutative. If we trace the number 11 around the diagram, it maps to 00 along the diagonal arrow, but it maps to 11 itself if we take the horizontal-then-vertical route. And 0≠10≠1. So to indicate if/when a given diagram is commutative, we have to say it explicitly. Or sometimes folks will use the symbols shown below to indicate commutativity:

‍

I think now is a good time to decode another phrase that often accompanies the commutative-diagram parlance. Returning to our f,g,hf,g,h example, we assumed that f,gf,g and h=g∘fh=g∘f already existed. But suppose we only knew about the existence of f:A→Bf:A→B and some other map, say, z:A→Cz:A→C. Then we might like to know, “Does there exist a map g:B→Cg:B→C such that z=g∘fz=g∘f? Perhaps the answer is no. Or perhaps the answer is yes, but only under certain hypotheses.* Well, if such a gg does exists, then we’ll say “…there exists a map gg such that the following diagram commutes:

but folks might also say

“…there exists a map gg such that zz factors through gg”

The word “factors” means just what you think it means. The diagram commutes if and only if z=g∘fz=g∘f, and that notation suggests that we may think of gg as a factor of zz, analogous to how 22 is a factor of 66 since 6=2⋅36=2⋅3.

By the way, we’ve only chatted about sets and functions so far, but diagrams make sense in any context in which you have mathematical objects and arrows. So we can talk about diagrams of groups and group homomorphisms, or vector spaces and linear transformations, or topological spaces and continuous maps, or smooth manifolds and smooth maps, or functors and natural transformations and so on. Diagrams make sense in any category. And as you can imagine, there are more complicated diagrams than triangular ones. For instance, suppose we have two more maps i:A→Di:A→D and j:D→Cj:D→C such that hh is equal to not only g∘fg∘f but also j∘ij∘i. Then we can express the equality g∘f=h=j∘ig∘f=h=j∘i by a square:

Again, commutativity simply tells us that the three ways of getting from AA to CC are all equivalent. And diagrams can get really crazy and involve other shapes too. They can even be three-dimensional! Here are some possibilities where I’ve used bullets in lieu of letters for the source and target of the arrows.

No matter the shape, the idea is the same: Any map can be thought of as a path or a process from AA to BB, from start to finish. And we use diagrams to capitalize on that by literally writing down “AA” and “BB” (or “∙∙” and “∙∙”) and by literally drawing a path—in the form of an arrow—between them.

 

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

 

I am often asked the question, “How do you do it?!” Now while I don’t think my note-taking strategy is particularly special, I am happy to share! I’ll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to “stick.” So while some people think aloud (or quietly),

I think on paper.

My study habits, then, are built on this fact. Of course not everyone learns in this way, so this post is not intended to be a how-to guide. It’s just a here’s-what-I-do guide.

With that said, below is a step-by-step process I tried to follow during my final years of undergrad and first two years of grad school.**

‍Step 1

Read the appropriate chapter/section in the book before class

I am an “active reader,” so my books have tons of scribbles, underlines, questions, and “aha” moments written on the pages. I like to write while I read because it gives me time to pause and think about the material. For me, reading a mathematical text is not like reading a novel. It often takes me a long time just to understand a single paragraph! Or a single sentence. I also like to mark things that I don’t understand so I’ll know what to look for in the upcoming lecture.

STEP 2

Attend lecture and take notes

This step is pretty self-explanatory, but I will mention this: I write down much more than what is written on the chalkboard (or whiteboard). In fact, a good portion of my in-class notes consists of what the professor has said but hasn’t written.

‍My arsenal

‍STEP 3

Rewrite lecture notes at home

My in-class notes are often an incomprehensible mess of frantically-scribbled hieroglyphs. So when I go home, I like to rewrite everything in a more organized fashion. This gives the information time to simmer and marinate in my brain. I’m able to ponder each statement at my own pace, fill in any gaps, and/or work through any exercises the professor might have suggested. I’ll also refer back to the textbook as needed.

Sometimes while rewriting these notes, I’ll copy things word-for-word (either from the lecture, the textbook, or both), especially if the material is very new or very dense. Although this can be redundant, it helps me slow down and lets me think about what the ideas really mean. Other times I’ll just rewrite things in my own words in a way that makes sense to me.

A semester’s worth of notes!

 

As for the content itself, my notes usually follow a “definition then theorem then proof” outline, simply because that’s how material is often presented in the lecture. But sometimes it’s hard to see the forest for the trees (i.e. it’s easy to get lost in the details), so I’ll occasionally write “PAUSE!” or “KEY IDEA!” in the middle of the page. I’ll then take the time to write a mini exposition that summarizes the main idea of the previous pages. I’ve found this to be especially helpful when looking back at my notes after several months (or years) have gone by. I may not have time to read all the details/calculations, so it’s nice to glance at a summary for a quick refresher.

And every now and then, I’ll rewrite my rewritten notes in the form of a SaiBlog post! Many of my earlier posts here at Math3ma were “aha” moments that are now engrained in my brain because I took the time to SaiBlog about them.

STEP 4

Do homework problems

Once upon a time, I used to think the following:

How can I do problems if I haven’t spent a bajillion hours learning the theory first?

But now I believe there’s something to be said for the converse: 

How can I understand the theory if I haven’t done a bajillion examples first?

In other words, taking good notes and understanding theory is one thing, but putting that theory into practice is a completely different beast. As a wise person once said, “The only way to learn math is to DO math.” So although I’ve listed “do homework problems” as the last step, I think it’s really first in terms of priority.

Typically, then, I’ll make a short to-do list (which includes homework assignments along with other study-related duties) each morning. And I’ll give myself a time limit for each task. For example, something like “geometry HW, 3 hours” might appear on my list, whereas “do geometry today” will not. Setting a time gives me a goal to reach for which helps me stay focused. And I may be tricking my brain here, but a specific, three-hour assignment sounds much less daunting than an unspecified, all-day task. (Of course, my lists always contain multiple items that take several hours each, but as the old adage goes, “How do you eat an elephant? One bite at a time.”)

By the way, in my first two years of grad school I often worked with my classmates on homework problems. I didn’t do this in college, but in grad school I’ve found it tricky to digest all the material alone – there’s just too much of it! So typically I’d first attempt exercises on my own, then meet up with a classmate or two to discuss our ideas and solutions and perhaps attend office hours with any questions.

As far as storage goes, I have a huge binder that contains all of my rewritten notes*** from my first and second year classes. (I use sheet protectors to keep them organized according to subject.) On the other hard, I use a paper tray like this one to store my lecture notes while the semester is in progress. Once classes are over, I’ll scan and save them to an external hard drive. I’ve also scanned and saved all my homework assignments.

Well, I think that’s about it! As I mentioned earlier, these steps were only my ideal plan. I often couldn’t apply them to every class — there’s just not enough time! — so I’d only do it for my more difficult courses. And even then, there might not be enough time for steps 1 and 3, and I’d have to start working on homework right after a lecture.

But as my advisor recently told me,”It’s okay to not know everything.” Indeed, I think the main thing is to just do something. Anything. As much as you can. And as time goes on, you realize you really are learning something, even if it doesn’t feel like it at the time.

Alright, friends, I think that’s all I have to share. I hope it was somewhat informative. If you have any questions, don’t hesitate to leave it in a comment below!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Mathematicians are stunned by the discovery that prime numbers are pickier than previously thought. The find suggests number theorists need to be a little more careful when exploring the vast infinity of primes.

Primes, the numbers divisible only by themselves and 1, are the building blocks from which the rest of the number line is constructed, as all other numbers are created by multiplying primes together. That makes deciphering their mysteries key to understanding the fundamentals of arithmetic.

Although whether a number is prime or not is pre-determined, mathematicians don’t have a way to predict which numbers are prime, and so tend to treat them as if they occur randomly. Now Kannan Soundararajan and Robert Lemke Oliver of Stanford University in California have discovered that isn’t quite right.

“It was very weird,” says Soundararajan. “It’s like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you’ve never seen before.”

Surprising order

So just what has got mathematicians spooked? Apart from 2 and 5, all prime numbers end in 1, 3, 7 or 9 – they have to, else they would be divisible by 2 or 5 – and each of the four endings is equally likely. But while searching through the primes, the pair noticed that primes ending in 1 were less likely to be followed by another prime ending in 1. That shouldn’t happen if the primes were truly random – consecutive primes shouldn’t care about their neighbour’s digits.

“In ignorance, we thought things would be roughly equal,” says Andrew Granville of the University of Montreal, Canada. “One certainly believed that in a question like this we had a very strong understanding of what was going on.”

The pair found that in the first hundred million primes, a prime ending in 1 is followed by another ending in 1 just 18.5 per cent of the time. If the primes were distributed randomly, you’d expect to see two 1s next to each other 25 per cent of the time. Primes ending in 3 and 7 take up the slack, each following a 1 in 30 per cent of primes, while a 9 follows a 1 in around 22 per cent of occurrences.

Similar patterns showed up for the other combinations of endings, all deviating from the expected random values. The pair also found them in other bases, where numbers are counted in units other than 10s. That means the patterns aren’t a result of our base-10 numbering system, but something inherent to the primes themselves. The patterns become more in line with randomness as you count higher – the pair have checked up to a few trillion – but still persists.

“I was very surprised,” says James Maynard of the University of Oxford, UK, who on hearing of the work immediately performed his own calculations to check the pattern was there. “I somehow needed to see it for myself to really believe it.”

Stretching to infinity

Thankfully, Soundararajan and Lemke Oliver think they have an explanation. Much of the modern research into primes is underpinned G H Hardy and John Littlewood, two mathematicians who worked together at the University of Cambridge in the early 20^th century. They came up with a way to estimate how often pairs, triples and larger grouping of primes will appear, known as the k-tuple conjecture.

Just as Einstein’s theory of relativity is an advance on Newton’s theory of gravity, the Hardy-Littlewood conjecture is essentially a more complicated version of the assumption that primes are random – and this latest find demonstrates how the two assumptions differ. “Mathematicians go around assuming primes are random, and 99 per cent of the time this is correct, but you need to remember the 1 per cent of the time it isn’t,” says Maynard.

The pair used Hardy and Littlewood’s work to show that the groupings given by the conjecture are responsible for introducing this last-digit pattern, as they place restrictions on where the last digit of each prime can fall. What’s more, as the primes stretch to infinity, they do eventually shake off the pattern and give the random distribution mathematicians are used to expecting.

“Our initial thought was if there was an explanation to be found, we have to find it using the k-tuple conjecture,” says Soundararajan. “We felt that we would be able to understand it, but it was a real puzzle to figure out.”

The k-tuple conjecture is yet to be proven, but mathematicians strongly suspect it is correct because it is so useful in predicting the behaviour of the primes. “It is the most accurate conjecture we have, it passes every single test with flying colours,” says Maynard. “If anything I view this result as even more confirmation of the k-tuple conjecture.”

Although the new result won’t have any immediate applications to long-standing problems about primes like the twin-prime conjecture or the Riemann hypothesis, it has given the field a bit of a shake-up. “It gives us more of an understanding, every little bit helps,” says Granville. “If what you take for granted is wrong, that makes you rethink some other things you know.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Yesterday I received a disheartening 44/50 on a homework assignment. Okay okay, I know. 88% isn’t bad, but I had turned in my solutions with so much confidence that admittedly, my heart dropped a little (okay, a lot!) when I received the grade. But I quickly had to remind myself, Hey! Grades don’t matter.

The six points were deducted from two problems. (Okay, fine. It was three. But in the third I simply made an air-brained mistake.) In the first, apparently my answer wasn’t explicit enough. How stingy! I thought. Doesn’t our professor know that this is a standard example from the book? I could solve it in my sleep! But after the prof went over his solution in class, I realized that in all my smugness I never actually understood the nuances of the problem. Oops. You bet I’ll be reviewing his solution again. Lesson learned.

In the second, I had written down my solution in the days before and had checked with a classmate and (yes) the internet to see if I was correct. Unfortunately, the odds were against me two-to-one as both sources agreed with each other but not with me. But I just couldn’t see how I could possibly be wrong! Confident that my errors were truths, I submitted my solution anyway, hoping there would be no consequences. But alas, points were taken off.

Honestly though, is a lower grade such a bad thing? I think not. In both cases, I learned exactly where my understanding of the material went awry. And that’s great! It means that my comprehension of the math is clearer now than it was before (and that the chances of passing my third qualifying exam have just increased. Woo!) And that’s precisely why I’m (still, heh…) in school.

So yes, contrary to what the comic above says, grades do exist in grad school, but – and this is what I think the comic is hinting at – they don’t matter. Your thesis committee members aren’t going to say, “Look, your defense was great, but we can’t grant you your PhD. Remember that one homework/midterm/final grade from three years ago?” (They may not use the word “great” either, but that’s another matter.) Of course, we students should still work hard and put in maximum effort! But the emphasis should not be on how well we perform, but rather how much we learn. Focus on the latter and the former will take care of itself. This is true in both graduate school and college, but the lack of emphasis on grades in grad school really brings it home. And personally, I’m very grateful for it because my brain is freed up to focus on other things like, I don’t know, learning math!

So to all my future imperfect homework scores out there: bring it on.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

In sum, the sufficient condition (a.k.a. the “if” direction) allows you to get what you want. That is, if you assume the sufficient condition, you’ll obtain your desired conclusion. It’s enough. It’s sufficient.

On the other hand, the necessary condition (a.k.a. the “only if” direction) is the one you must assume in order to get what you want. In other words, if you don’t have the necessary condition then you can’t reach your desired conclusion. It is necessary.

Here’s a little graphic which summarizes this:

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Here’s one thing to impress your friends with the next time you order a takeaway: new and exotic ways to slice a pizza.

Most of us divide a pizza using straight cuts that all meet in the middle. But what if the centre of the pizza has a topping that some people would rather avoid, while others desperately want crust for dipping?

Mathematicians had previously come up with a recipe for slicing – formally known as a monohedral disc tiling – that gives you 12 identically shaped pieces, six of which form a star extending out from the centre, while the other six divide up the crusty remainder. You start by cutting curved three-sided slices across the pizza, then dividing these slices in two to get the inside and outside groups, as shown below.

Now Joel Haddley and Stephen Worsley of the University of Liverpool, UK, have generalised the technique to create even more ways to slice. The pair have proved you can create similar tilings from curved pieces with any odd number of sides – known as 5-gons, 7-gons and so on (shaded below) – then dividing them in two as before. “Mathematically there is no limit whatsoever,” says Haddley, though you might find it impractical to carry out the scheme beyond 9-gon pieces.

Haddley and Worsley went one further by cutting wedges in the corners of their shapes, creating bizarre, spikey pieces that still form a circle (the image below shows this happening with 5-gons). “It’s really surprising,” says Haddley.

 

As with many mathematical results, its usefulness isn’t immediately obvious. The same is true of another pizza theorem, which looks at what happens when a pizza is haphazardly cut off-centre.

“I’ve no idea whether there are any applications at all to our work outside of pizza-cutting,” says Haddley, who has actually tried slicing a pizza in this way for real (see below). But the results are “interesting mathematically, and you can produce some nice pictures”.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

A maths problem previously tackled with the help of a computer, which produced a proof the size of Wikipedia, has now been cut down to size by a human. Although it is unlikely to have practical applications, the result highlights the differences between two modern approaches to mathematics: crowdsourcing and computers.

Terence Tao of the University of California, Los Angeles, has published a proof of the Erdős discrepancy problem, a puzzle about the properties of an infinite, random sequence of +1s and -1s. In the 1930s, Hungarian mathematician Paul Erdős wondered whether such a sequence would always contain patterns and structure within the randomness.

One way to measure this is by calculating a value known as the discrepancy. This involves adding up all the +1s and -1s within every possible sub-sequence. You might think the pluses and minuses would cancel out to make zero, but Erdős said that as your sub-sequences got longer, this sum would have to go up, revealing an unavoidable structure. In fact, he said the discrepancy would be infinite, meaning you would have to add forever, so mathematicians started by looking at smaller cases in the hopes of finding clues to attack the problem in a different way.

Last year, Alexei Lisitsa and Boris Konev of the University of Liverpool, UK used a computer to prove that the discrepancy will always be larger than two. The resulting proof was a 13 gigabyte file – around the size of the entire text of Wikipedia – that no human could ever hope to check.

Helping hands

Tao has used more traditional mathematics to prove that Erdős was right, and the discrepancy is infinite no matter the sequence you choose. He did it by combining recent results in number theory with some earlier, crowdsourced work.

In 2010, a group of mathematicians, including Tao, decided to work on the problem as the fifth Polymath project, an initiative that allows professionals and amateurs alike to contribute ideas through SaiBlogs and wikis as part of mathematical super-brain. They made some progress, but ultimately had to give up.

“We had figured out an interesting reduction of the Erdős discrepancy problem to a seemingly simpler problem involving a special type of sequence called a completely multiplicative function,” says Tao.

Then, in January this year, a new development in the study of these functions made Tao look again at the Erdős discrepancy problem, after a commenter on his SaiBlog pointed out a possible link to the Polymath project and another problem called the Elliot conjecture.

Not just conjecture

“At first I thought the similarity was only superficial, but after thinking about it more carefully, and revisiting some of the previous partial results from Polymath5, I realised there was a link: if one could prove the Elliott conjecture completely, then one could also resolve the Erdős discrepancy problem,” says Tao.

“I have always felt that that project, despite not solving the problem, was a distinct success,” writes University of Cambridge mathematician Tim Gowers, who started the Polymath project and hopes that others will be encouraged to participate in future. “We now know that Polymath5 has accelerated the solution of a famous open problem.”

Lisitsa praises Tao for doing what his algorithm couldn’t. “It is a typical example of high-class human mathematics,” he says. But mathematicians are increasingly turning to machines for help, a trend that seems likely to continue. “Computers are not needed for this problem to be solved, but I believe they may be useful in other problems,” Lisitsa says.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

 

Felker prefaces the quote by saying,

Giving up on math means you don’t believe that careful study can change the way you think.

He further notes that writing, like math, “is also not something that anyone is ‘good’ at without a lot of practice, but it would be completely unacceptable to think that your composition skills could not improve.”

Friends, this is so true! Being ‘good’ at math boils down to hard work and perseverance, not whether or not you have the ‘math gene.’ “But,” you might protest, “I’m so much slower than my classmates are!” or “My educational background isn’t as solid as other students’!” or “I got a late start in mathematics!”* That’s okay! A strong work ethic and a love and enthusiasm for learning math can shore up all deficiencies you might think you have. Now don’t get me wrong. I’m not claiming it’ll be a walk in the park. To be honest, some days it feels like a walk through an unfamiliar alley at nighttime during a thunderstorm with no umbrella. But, you see, that’s okay too. It may take some time and the road may be occasionally bumpy, but it can be done!

This brings me to another point that Felker makes: If you enjoy math but find it to be a struggle, do not be discouraged! The field of math is HUGE and its subfields come in many different flavors. So for instance, if you want to be a math major but find your calculus classes to be a challenge, do not give up! This is not an indication that you’ll do poorly in more advanced math courses. In fact, upper level math classes have a completely (I repeat, completely!) different flavor than calculus. Likewise, in graduate school you may struggle with one course, say algebraic topology, but find another, such as logic, to be a breeze. Case in point: I loathed real analysis as an undergraduate** and always thought it was pretty masochistic. But real analysis in graduate school was nothing like undergraduate real analysis (which was more like advanced calculus), and now – dare I say it? – I sort of enjoy the subject. (Gasp!)

All this to say that although Felker’s article is aimed at folks who may be afraid to take college-level math, I think it applies to math majors and graduate students too. I highly recommend you read it if you ever need a good ‘pick-me-up.’ And on those days when you feel like the math struggle is harder than usual, just remember:

Even the most accomplished mathematicians had to learn HOW to learn this stuff!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*