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Counting on maths

One team’s mission to raise humanity’s next-generation problem solvers

By Tom Almeroth-Williams

Image: skynesher

The world needs more people who can think mathematically to solve its mounting problems.

This is what drives the researchers behind nrich , cambridge’s flagship maths outreach project..

Now celebrating its 25 th anniversary, NRICH spent the last two years in unexpected emergency rescue mode, helping learners in Covid-19 lockdown.

The team continues to support the post-pandemic catch-up effort but Director Dr Ems Lord also has her sights set on big plans for the future.

“Our online resources were getting over a million page views per week,” Lord recalls. “ We were ready from day one of lockdown, and the huge response showed just how much that was needed.”

In Spring 2020, the UK Government highlighted NRICH resources to schools and the team contributed to the BBC’s heavily used Bitesize maths resources.

Between March and September 2020, nrich.maths.org registered a 95% increase in UK visits compared to the previous year. In the 2020–21 school year alone, the site attracted just under 33 million page views.

“We were about to publish new resources for schools which had been a huge amount of work,” Lord recalls. “We just dropped that and started gathering all the resources we knew families could do at home with buttons, counters, beans, whatever they had lying around, plus all the interactive games we had.”

Maths at Home catered, and still does, for every key stage from early years to sixth form and went live the very day that England sent its schoolchildren home. The following Monday, visits to the NRICH website immediately surged.

Professor Colm-Cille Caulfield, head of Cambridge’s department of applied maths and theoretical physics, says:

“The NRICH team had all the right skills, ideas and resources to respond to the pandemic. They pivoted amazingly to home learning and sorted out a huge amount of support for people across the globe. It was amazing to watch them do it.”

Dr Ems Lord, Director of NRICH. Image: Nathan Pitt

NRICH at 25 film:

The complete package, nrich, so-called because it seeks to enrich the teaching and learning of mathematics, has designed thousands of online resources for every stage of early years, primary and secondary school education..

Based at Cambridge’s Faculty of Mathematics, the project has always shared its materials online, free of charge and without any barriers, internationally. Its current team of seven includes a researcher, experienced teachers and web developers.

NRICH focuses on building problem-solving skills, perseverance, mathematical reasoning, ability to apply knowledge creatively in unfamiliar contexts, and confidence in tackling new challenges.

The team designs what it calls 'low threshold high ceiling' resources to be accessible to all learners, encourage exploration and discussion.

NRICH has always had a penchant for fun titles and real-world set-ups, as in Dodgy proofs ; Olympic Torch Tour , Wipeout ; Egyptian rope ; National flags ; and Scratch cards .

In addition to its brimming reservoir of resources, the team posts live problems so students can share their solutions for publication on the site.

Recently, the team posted solutions to an odd numbers problem from a student at Buckler's Mead Academy in Somerset, a British School in Oman, and the London Academy in Morocco. Teachers also regularly Tweet about using NRICH in the classroom.

Well done, Y2s! We're delighted that you enjoyed exploring this challenge, thanks for sharing your photos with our team. PS If other classes would like to explore Chain of Changes too, here's the task, teacher notes and more examples of classroom work https://t.co/bZirUPGhJb https://t.co/gDhfxagrlL — NRICH maths (@nrichmaths) March 9, 2022

Students are encouraged to visit the website themselves at home, not least during lockdowns, but NRICH has always seen working with teachers as the most effective and efficient way to serve as many young people as possible.

For this reason, the team provides specialist in-person and webinar training, plus resource-specific notes, for teachers to help them build lessons around ready-to-go tasks and maximise their impact in the classroom. The team also organises webinars for thousands of students from across the world.

Poster for a problem aimed at 11 – 16 year olds

Education, education, education

Remember that speech well, back in 1996, around the time tony blair was declaring these priorities for government, researchers at cambridge’s education faculty were brainstorming what would become nrich..

Ems Lord explains: “There's a long history of people sharing maths problems and solutions with each other, but that used to be slow and small-scale. When the internet arrived, colleagues at Education decided to create an online maths club.”

“Meanwhile, researchers here at Maths were thinking about how to mark the new millennium with a public outreach project. It made sense to bring both together so today NRICH is at the heart of The Millennium Mathematics Project .”

NRICH remains a collaboration between the two faculties. Lord took charge in 2015, following a career as a maths teacher, local authority consultant and head of one of the country's largest maths teacher training professional development programmes.

She was president of the Mathematical Association in 2019–20 and is a regular contributor to the All-Party Parliamentary Group for the Teaching Profession, as well as being a member of the Joint Mathematical Council and a founding fellow of the Chartered College of Teachers. 

Complementing her work at NRICH, Lord uses her research fellowship at Clare Hall to identify ways to improve student outcomes, address the gender imbalance in mathematics study post-16 and widen participation in the subject.

Lord says: “As many children get older, their curiosity often dips. And in this country especially, there are pervasive negative attitudes towards maths, so it's important that we catch people young and sustain their interest.”

“It's impossible to separate mathematics from imagination.”

Dr Ems Lord

“How do you have a breakthrough if somebody isn't curious and thinking, ‘I wonder if that's where the magic happens?’ Imagination is how we go from just solving problems to posing even more ambitious questions and moving forwards.

“What's really special about NRICH is that we encourage children to problem solve in different ways, take different approaches, and use and apply their maths. And by doing that, when students leave school, they are truly ready to solve problems.”

To take just one example, in Tree tops , a challenge aimed at 14–16 year olds, NRICH users are invited to enter the forestry business and work out the maximum profit that they could make after 100 years.

Curriculum cubed

In 1996, tony blair complained that the uk was, “35th in the world league of education standards … they say give me the boy at seven, i’ll show you the man at 70. well give me the education system that is 35th in the world today and i will give you the economy that is 35th in the world tomorrow.”.

There have been significant signs of improvement since then, not least in mathematics. In 2019, the Pisa tests, run by the Organisation for Economic Co-operation and Development, ranked the country 18 th for mathematics.

“The UK education system has done fantastic things,” says Lord, “but big challenges remain.”

She points out that while maths is now the most popular A-Level subject, only 6% of maths professors in the UK are female. “We still have a lot of work to do on that front. We need to ensure that the subject reflects the country’s diversity in lots of other ways too.”

Teamwork at an NRICH resilience workshop. Image: MMP/Nigel Luckhurst

NRICH maps onto the national curriculum for mathematics, which currently has three major strands: fluency, reasoning and problem solving. But even Ofsted, the school inspectors, admit that there isn’t enough emphasis on problem solving. If this is going to change, schools, students and policy makers will need to make even greater use of NRICH.

“Yes, we can improve exam performance,” says Lord, “ but our ultimate goal is to help students to think flexibly and solve problems.”

“In the UK, assessments at primary and secondary schools focus on number skills. These are important, of course. We need to know when our calculations are accurate, but we also need to value problem solving because that's what we face in everyday life. That isn’t evident in our exams yet.”

“There’s a difference between passing an exam and thinking like a mathematician. NRICH focuses on the latter.”

Ems Lord, Director of NRICH

At the same time, Lord thinks that the UK curriculum needs to put greater emphasis on developing collaborative skills, something which she has discussed with the All-Party Parliamentary Group for the Teaching Profession.

Lord remembers a six-year-old telling her that she couldn’t work on a maths task with someone else because it would be cheating.

“I was horrified,” she says. "Solving problems together is crucial. If you want proof of that just walk around our faculty and you’ll find some of the greatest mathematicians in the world working in teams all the time.”

Teacher’s report

Mark dawes , an experienced maths teacher at comberton village college in cambridgeshire has been using nrich ever since it was launched..

“A problem is, by definition, difficult,” he says, “so having high-quality problems that are really interesting, that create opportunities to try things out, and encourage pupils to share their ideas, that’s extremely powerful. As teachers, we just don't have time to sit down every week to devise new problems of this quality.

“Even when I’ve already solved the problems myself, when I teach with an NRICH resource, my pupils almost always come up with something that I hadn't thought of, and that democratises the classroom.”

“There are jobs that require a very deep mathematical understanding, but not everyone is going to become a computer programmer. What we do all need, however, is to be able to critically analyse what’s going on around us, to make decisions based on data and reasoning.

“NRICH helps pupils to use the full range of their maths skills and opportunities to apply them. And it shows them how useful maths can be in so many different ways.”

Image captions.

Going forwards, Ems Lord hopes to work even more closely with government to ensure that future generations have the problem solving skills that they will need.

She says: “Whether you’re fighting the spread of infectious diseases, trying to prevent climate change or adapt to increasing automation at work, you need people with curious minds who can work flexibly, make connections and collaborate. And that's why NRICH is so important.”

Professor Caulfield agrees: “We need to embed mathematics, and an understanding of how it plays a role in absolutely everything, in our education system.

“NRICH teaches people to think like mathematicians and learn how to solve problems, the kind of problem-solving that will allow us to have a sustainable and healthy society.”

Professor Colm-Cille Caulfield

“As the internet and all of its data become more important in our lives, we have to adapt to cope with that.

“Understanding how mathematics underpins data and how you use it to solve problems is exactly what NRICH communicates to every child.”

Julia Gog, Professor of mathematical biology at Cambridge and another long-term NRICH supporter, says:

“The pandemic has shown that mathematical literacy for everyone is more important than ever.

“The more you have, the more you can make sense of what is happening around you, ask the right questions and make informed personal choices.”

Ems Lord sees an exciting future for NRICH. In the medium term, the team will be working to make it even easier for teachers to deliver sequences of lessons which develop and embed key transferable skills among their students.

Longer term, they hope to harness advances in AI, and adapt work already being done in the Maths Faculty on medical imaging, to make NRICH more responsive to how individual students think.

Lord says: “When a student gets stuck, we want to our resources to be capable of prompting them with personalised next steps, offering different visuals and approaches. This will take time, but we need to aim high and tap in to the research happening around us.”

...visit our interactive map to find out how.

Find out how you can support NRICH here .

Professor Julia Gog

Explore nrich.maths.org

Published 30 March 2022 With thanks to:

All of the outstanding people who have made NRICH such a success over the past 25 years

Mark Dawes and Comberton Village College

Professors Colm-Cille Caulfield and Julia Gog

Words: Tom Almeroth-Williams

Film by Jonathan Settle. Story designed with Alison Fair

The text in this work is licensed under a Creative Commons Attribution 4.0 International License  

7 of the hardest problems in mathematics that have been solved

Mathematics is full of problems, some of which have been solved and others that haven’t. here, we focus on 7 of the hardest problems that have been solved..

Tejasri Gururaj

What are the hardest problems in math that have already been solved?

intararit  

• Some problems in mathematics have taken centuries to be solved, due to their complexity.
• Although there are some complex math problems that still elude solutions, others have now been solved.
• Here are 7 of the hardest math problems ever solved.

Some mathematical problems are challenging even for the most accomplished mathematicians.

From the Poincaré conjecture to Fermat’s last theorem , here we take a look at some of the most challenging math problems ever solved.

1. Poincaré conjecture

Salix alba  

The Poincaré conjecture is a famous problem in topology, initially proposed by French mathematician and theoretical physicist Henri Poincaré in 1904. It asserts that every simply connected, closed 3-manifold is topologically homeomorphic (a function that is a one-to-one mapping between sets such that both the function and its inverse are continuous) to a 3-dimensional sphere. 

In simpler terms, the conjecture asserts that a particular group of three-dimensional shapes can be continuously transformed into a sphere without any gaps or holes.

The problem was solved by the reclusive Russian mathematician Grigori Perelman in 2003. He built upon the work of American mathematician Richard S. Hamilton’s program involving the Ricci flow.

What makes this achievement even more remarkable is that Perelman declined the prestigious Fields Medal and the Clay Millennium Prize reward that came with it. He chose to stay away from the spotlight and mathematical acclaim, but his proof withstood rigorous scrutiny from the mathematical community.

The resolution of the Poincaré conjecture confirmed the fundamental role of topology in understanding the shape and structure of spaces, impacting fields like geometry and manifold theory.

2. The prime number theorem 

Britannica  

The prime number theorem long stood as one of the fundamental questions in number theory. At its core, this problem is concerned with unraveling the distribution of prime numbers. 

The question at hand revolves around the distribution pattern of these primes within the realm of natural numbers. Are there any discernible patterns governing the distribution of prime numbers, or do they appear to be entirely random? The theorem states that for large values of x, π(x) is approximately equal to x/ln(x).

The breakthrough in solving this theorem came in the late 19th century, thanks to the independent work of two mathematicians, Jacques Hadamard and Charles de la Vallée-Poussin. In 1896, both mathematicians presented their proofs of the theorem. 

Their work demonstrated that prime numbers exhibit a remarkable, asymptotic distribution pattern . Their solution produced the insight that, as one considers larger and larger numbers, the density of prime numbers diminishes. 

The theorem precisely characterized the rate of this decrease, showing that prime numbers become less frequent as we move along the number line. It’s as if they gradually thin out, although they never entirely vanish.

The prime number theorem was a turning point in the study of number theory . It provided profound insights into the distribution of prime numbers and put to rest the notion that there might exist a formula predicting each prime number. 

Instead, it proposed a probabilistic approach to understanding the distribution of prime numbers. The theorem’s significance extends into various mathematical fields, especially in cryptography, where the properties of prime numbers play a pivotal role in securing communications.

3. Fermat’s last theorem

Charles Rex Arbogast/AP via NPR  

Fermat’s last theorem is one of the problems on this list many people are most likely to have heard of. The conjecture, proposed by French mathematician Pierre de Fermat in the 17th century, states that it’s impossible to find three positive integers, a, b, and c, that can satisfy the equation a n + b n = c n for any integer value of n greater than 2.

For instance, there are no whole number values of a, b, and c that can make 3 3 + 4 3 = 5 3 true.

This problem remained unsolved for centuries and became one of the most challenging problems in mathematics. It was made more enticing because Fermat apparently wrote a note in his copy of the  Arithmetica  by Diophantus of Alexandria, saying, “I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.”

Numerous mathematicians attempted to prove or disprove Fermat’s conjecture, but it wasn’t until 1994 that the breakthrough came.

The solution was achieved by British mathematician Andrew Wiles, who built upon the work of many other mathematicians who had contributed to the field of number theory. Wiles’ proof was extraordinarily complex and required intricate mathematical concepts and theorems, particularly those related to elliptic curves and modular forms.

Wiles’ remarkable proof of Fermat’s last theorem confirmed that the conjecture was indeed true. Having taken more than three centuries to be solved, it had a profound impact on the world of mathematics, demonstrating the power of advanced mathematical techniques in solving long-standing problems.

Z. Ziegler, M. Ondrachek  

Before Wiles presented its proof, it was in the Guinness Book of World Records as the “most difficult mathematical problem,” in part because the theorem has seen the greatest number of unsuccessful proofs.

4. Classification of finite simple groups

Jakob.scholbach/Pbroks13  

This one is a bit different from the others on the list. The classification of finite simple groups , also known as the “enormous theorem,” set out to classify all finite simple groups, which are the fundamental building blocks of group theory.

Finite simple groups are those groups that cannot be divided into smaller non-trivial normal subgroups. The goal was to understand and categorize all the different types of finite simple groups that exist.

The solution to this problem is not straightforward. The proof is a collaborative effort by hundreds of mathematicians covering tens of thousands of pages in hundreds of journal articles published between 1955 and 2004.

It is one of the most extensive mathematical proofs ever produced and marks a monumental achievement in group theory.

The proof outlines the structure of finite simple groups and demonstrates that they can be classified into several specific categories. This achievement paved the way for a deeper understanding of group theory and its applications in various mathematical fields. 

5. The four color theorem

Inductiveload  

The four color theorem tackles an intriguing question related to topology and stands as one of the first significant theorems proved by a computer. 

It states that any map in a plane can be colored using four colors so that no two adjacent regions share the same color while using the fewest possible colors. Adjacent, in this context, means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.

The theorem doesn’t focus on the artistic aspect of map coloring but rather on the fundamental mathematical principles that underlie it.

The solution to this theorem arrived in 1976, thanks to the combined efforts of mathematicians Kenneth Appel and Wolfgang Haken. However, the proof was not widely accepted due to the infeasibility of checking it by hand.

Appel and Haken’s achievement confirmed that any map, regardless of its complexity, can be colored with just four colors such that no two neighboring regions share the same color. While the idea seems simple, proving it rigorously was complex and time-consuming.

To address any lingering skepticism about the Appel–Haken proof, a more accessible proof using similar principles and still utilizing computer assistance was presented in 1997 by Robertson, Sanders, Seymour, and Thomas. 

Additionally, in 2005, Georges Gonthier achieved a proof of the theorem using general-purpose theorem-proving software, reinforcing the credibility of the four color theorem.

This theorem is not actually used in map-making but has far-reaching implications in various fields, from graph theory to computer science, where it finds applications in scheduling, circuit design, and optimization problems.

6. Gödel’s incompleteness theorems

Andrew Das Arulsamy/Research Gate  

Gödel’s incompleteness theorems , formulated by Austrian mathematician Kurt Gödel in the 20th century, delve into the mysteries of formal systems and their inherent limitations. 

In mathematics, a formal system has a structured and well-defined framework or language that comprises a set of symbols, rules, and axioms employed for representing and manipulating mathematical or logical expressions.

Gödel’s first incompleteness theorem explores a fundamental question: In any consistent formal system, are there true mathematical statements that are undecidable within that system? In other words, do statements exist that cannot be proven as either true or false using the rules and axioms of that system?

The second incompleteness theorem takes this further: Can any consistent formal system prove its own consistency?

Gödel not only posed these questions but also provided the answers. He established, through rigorous mathematical proofs , that there exist true statements within formal systems that cannot be proven within those very systems.

In essence, the first theorem asserts that there are statements that cannot be proven as either true or false using the rules and axioms of a system. The second theorem demonstrates that no consistent formal system can prove its own consistency.

Gödel’s theorems introduced a profound paradox within the realm of mathematical logic: There are truths that exist beyond the reach of formal proofs, and there are limits to what can be achieved through mathematical systems alone.

Gödel’s contributions to mathematical logic influenced the philosophy of mathematics and our understanding of the inherent limits of formal systems.

7. The goat problem

Mnchnstnr  

The goat problem is a much more recently solved mathematical problem. It involves calculating the grazing area for a tethered goat. Despite its initial simplicity, mathematicians have pondered this problem for over a century.

In its basic form, a goat on a rope can graze in a semicircle with an area of A = 1/2πr 2 , where r is the rope’s length. However, the problem becomes more complex when you change the shape of the area the goat can access.

For instance, when tethered to a square barn, the goat can access more than just a semicircle. The goat can also go around the corners of the barn, creating additional quarter circles.

Mathematician Ingo Ullisch recently unraveled the goat grazing problem, introducing complex analysis into the equation. However, the solution is far from elementary.

It involves intricate calculations, relying on the ratio of contour integral expressions and involves numerous trigonometric terms . Although the solution may not offer a practical guide for goat owners, it represents a significant achievement in the world of mathematics.

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What makes the goat problem truly fascinating is its capacity to act as a mathematical Rosetta stone , transcending boundaries between various fields and serving as a versatile challenge for experts from diverse disciplines.

From age-old conundrums that took centuries to crack to enigmas that continue to elude solutions, mathematical mysteries remind us that the pursuit of knowledge is an ever-evolving journey.

So, the next time you find yourself pondering a difficult math problem, remember that you are in good company, following in the footsteps of the greatest mathematical explorers!

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ABOUT THE EDITOR

Tejasri Gururaj Tejasri is a versatile Science Writer & Communicator, leveraging her expertise from an MS in Physics to make science accessible to all. In her spare time, she enjoys spending quality time with her cats, indulging in TV shows, and rejuvenating through naps.

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Mathematicians Are Edging Close to Solving One of the World's 7 Hardest Math Problems

And there’s $1 million at stake.

• In new research, mathematicians have narrowed down one of the biggest outstanding problems in math.
• Huge breakthroughs in math and science are usually the work of many people over many years.
• Seven math problems were given a $1 million bounty each in 2000, and just one has been solved so far.

The “Millennium Problems” are seven infamously intractable math problems laid out in the year 2000 by the prestigious Clay Institute, each with $1 million attached as payment for a solution. They span all areas of math , as the Clay Institute was founded in 1998 to push the entire field forward with financial support for researchers and important breakthroughs.

But the only solved Millennium Problem so far, the Poincare conjecture, illustrates one of the funny pitfalls inherent to offering a large cash prize for math. The winner, Grigori Perelman, refused the Clay prize as well as the prestigious Fields Medal. He withdrew from mathematics and public life in 2006, and even in 2010, he still insisted his contribution was the same as the mathematician whose work laid the foundation on which he built his proof, Richard Hamilton.

Math, all sciences, and arguably all human inquiries are filled with pairs or groups that circle the same finding at the same time until one officially makes the breakthrough. Think about Sir Isaac Newton and Gottfried Leibniz, whose back-and-forth about calculus led to the combined version of the field we still study today. Rosalind Franklin is now mentioned in the same breath as her fellow discoverers of DNA, James Watson and Francis Crick. Even the Bechdel Test for women in media is sometimes called the Bechdel-Wallace Test, because humans are almost always in collaboration.

That’s what makes this new paper so important. Two mathematicians—Larry Guth of the Massachusetts Institute of Technology (MIT) and James Maynard of the University of Oxford—collaborated on the new finding about how certain polynomials are formed and how they reach out into the number line. Maynard is just 37, and won the Fields Medal himself in 2022. Guth, a decade older, has won a number of important prizes with a little less name recognition.

The Riemann hypothesis is not directly related to prime numbers , but it has implications that ripple through number theory in different ways (including with prime numbers). Basically, it deals with where and how the graph of a certain function of complex numbers crosses back and forth across axes. The points where the function crosses an axis is called a “zero,” and the frequency with which those zeroes appear is called the zero density.

In the far reaches of the number line, prime numbers become less and less predictable (in the proverbial sense). They are not, so far, predictable in the literal sense—a fact that is an underpinning of modern encryption , where data is protected by enormous strings of integers made by multiplying enormous prime numbers together. The idea of a periodic table of primes, of any kind of template that could help mathematicians better understand where and how large primes cluster together or not, is a holy grail.

In the new paper, Maynard and Guth focus on a new limitation of Dirichlet polynomials. These are special series of complex numbers that many believe are of the same type as the function involved in the Riemann hypothesis involves. In the paper, they claim they’ve proven that these polynomials have a certain number of large values, or solutions , within a tighter range than before.

In other words, if we knew there might be an estimated three Dirichlet values between 50 and 100 before, now we may know that range to be between 60 and 90 instead. The eye exam just switched a blurry plate for a slightly less blurry one, but we still haven’t found the perfect prescription. “If one knows some more structure about the set of large values of a Dirichlet polynomial, then one can hope to have improved bound,” Maynard and Guth conclude.

No, this is not a final proof of the Riemann hypothesis. But no one is suggesting it is. In advanced math, narrowing things down is also vital. Indeed, even finding out that a promising idea turns out to be wrong can have a lot of value—as it has a number of times in the related Twin Primes Conjecture that still eludes mathematicians.

In a collaboration that has lasted 160 years and counting, mathematicians continue to take each step together and then, hopefully, compare notes.

Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all. 

Number Line

The Algebra Calculator is a versatile online tool designed to simplify algebraic problem-solving for users of all levels. Here's how to make the most of it:

• Begin by typing your algebraic expression into the above input field, or scanning the problem with your camera.
• After entering the equation, click the 'Go' button to generate instant solutions.
• The calculator provides detailed step-by-step solutions, aiding in understanding the underlying concepts.
• -x+3\gt 2x+1
• (x+5)(x-5)\gt 0
• 10^{1-x}=10^4
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July 1, 2024

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.*

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

*Editor’s Note (7/9/24): This sentence was edited after posting to better clarify that prime numbers exclude 1.

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Solving word problems involving triangles and implications on training pre-service mathematics teachers

• William Guo , 
• School of Engineering and Technology, Central Queensland University, North Rockhampton, QLD 4702, Australia
• Academic Editor: Zlatko Jovanoski
• Received: 18 June 2024 Revised: 02 July 2024 Accepted: 05 July 2024 Published: 09 July 2024
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Triangles and trigonometry are always difficult topics for both mathematics students and teachers. Hence, students' performance in solving mathematical word problems in these topics is not only a reflection of their learning outcomes but also an indication of teaching effectiveness. This case study drew from two examples of solving word problems involving triangles by pre-service mathematics teachers in a foundation mathematics course delivered by the author. The focus of this case study was on reasoning implications of students' performances on the effective training of pre-service mathematics teachers, from which a three-step interactive explicit teaching-learning approach, comprising teacher-led precise and inspiring teaching (or explicit teaching), student-driven engaged learning (or imitative learning), and student-led and teacher-guided problem-solving for real-world problems or projects (or active application), was summarized. Explicit teaching establishes a solid foundation for students to further their understanding of new mathematical concepts and to conceptualize the technical processes associated with these new concepts. Imitative learning helps students build technical abilities and enhance technical efficacy by engaging in learning activities. Once these first two steps have been completed, students should have a decent understanding of new mathematical concepts and technical efficacy to analyze, formulate, and finally solve real-world applications with assistance from teachers whenever required. Specially crafted professional development should also be considered for some in-service mathematics teachers to adopt this three-step interactive teaching-learning process.

• pre-service mathematics teacher ,
• word problem ,
• triangles ,
• problem-solving ,
• explicit teaching ,
• imitated learning ,
• active applications ,
• professional development

Citation: William Guo. Solving word problems involving triangles and implications on training pre-service mathematics teachers[J]. STEM Education, 2024, 4(3): 263-281. doi: 10.3934/steme.2024016

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• This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ -->

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• Figure 1. A sketch of isosceles triangle for the first word problem
• Figure 2. A reworked sketch for the second problem with derived angles (in red)
• Figure 3. The first reworked sketch for solving the second problem through right triangles
• Figure 4. The second reworked sketch for solving the second problem through right triangles
• Figure 5. The third reworked sketch for solving the second problem through right triangles

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Title: solving for x and beyond: can large language models solve complex math problems with more-than-two unknowns.

Abstract: Large Language Models (LLMs) have demonstrated remarkable performance in solving math problems, a hallmark of human intelligence. Despite high success rates on current benchmarks; however, these often feature simple problems with only one or two unknowns, which do not sufficiently challenge their reasoning capacities. This paper introduces a novel benchmark, BeyondX, designed to address these limitations by incorporating problems with multiple unknowns. Recognizing the challenges in proposing multi-unknown problems from scratch, we developed BeyondX using an innovative automated pipeline that progressively increases complexity by expanding the number of unknowns in simpler problems. Empirical study on BeyondX reveals that the performance of existing LLMs, even those fine-tuned specifically on math tasks, significantly decreases as the number of unknowns increases - with a performance drop of up to 70\% observed in GPT-4. To tackle these challenges, we propose the Formulate-and-Solve strategy, a generalized prompting approach that effectively handles problems with an arbitrary number of unknowns. Our findings reveal that this strategy not only enhances LLM performance on the BeyondX benchmark but also provides deeper insights into the computational limits of LLMs when faced with more complex mathematical challenges.

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