Category: mathematics

Ten! Alamy/Insidefoto

Humans are finite creatures. Our brains have a finite number of neurons and we interact with a finite number of people during our finite lifetime. Yet humans have the remarkable ability to conceive of the infinite.

This ability underlies Euclid’s proof that there are infinite prime numbers as well as the belief of billions that their gods are infinite beings, free of mortal constraints.

These ideas will be well known to Pope Leo XIV since before his life in the church, he trained as a mathematician. Leo’s trajectory is probably no coincidence since there is a connection between mathematics and theology.

Infinity is undoubtedly of central importance to both. Virtually all mathematical objects, such as numbers or geometric shapes, form infinite collections. And theologians frequently describe God as a unique, absolutely infinite being.

Despite using the same word, though, there has traditionally been a vast gap between how mathematicians and theologians conceptualise infinity. From antiquity until the 19th century, mathematicians have believed that there are infinitely many numbers, but – in contrast to theologians – firmly rejected the idea of the absolute infinite.

The idea roughly is this: surely, there are infinitely many numbers, since we can always keep counting. But each number itself is finite – there are no infinite numbers. What is rejected is the legitimacy of the collection of all numbers as a closed object in its own right. For the existence of such a collection leads to logical paradoxes.

A paradox of the infinite

The most simple example is a version of Galileo’s paradox and leads to seemingly contradictory statements about the natural numbers 1,2,3….

First, observe that some numbers are even, while others are not. Hence, the numbers – even and odd – must be more numerous than just the even numbers 2,4,6…. And yet, for every number there is exactly one even number. To see this, simply multiply any given number by 2.

But then there cannot be more numbers than there are even numbers. We thus arrive at the contradictory conclusion that numbers are more numerous than the even numbers, while at the same time there are not more numbers than there are even numbers.

Because of such paradoxes, mathematicians rejected actual infinities for millennia. As a result, mathematics was concerned with a much tamer concept of infinity than the absolute one used by theologians. This situation dramatically changed with mathematician Georg Cantor’s introduction of transfinite set theory in the second half of the 19th century.

Georg Cantor, mathematical rebel. Wikipedia

Cantor’s radical idea was to introduce, in a mathematically rigorous way, absolute infinities to the realm of mathematics. This innovation revolutionised the field by delivering a powerful and unifying theory of the infinite. Today, set theory provides the foundations of mathematics, upon which all other subdisciplines are built.

According to Cantor’s theory, two sets – A and B – have the same size if their elements stand in a one-to-one correspondence. This means that each element of A can be related to a unique element of B, and vice versa.

Think of sets of husbands and wives respectively, in a heterosexual, monogamous society. These sets can be seen to have the same size, even though we might not be able to count each husband and wife.

The reason is that the relation of marriage is one-to-one. For each husband there is a unique wife, and conversely, for each wife there is a unique husband.

Using the same idea, we have seen above that in Cantor’s theory, the set of numbers – even and odd – has the same size as the set of even numbers. And so does the set of integers, which includes negative numbers, and the set of rational numbers, which can be written as fractions.

The most striking feature of Cantor’s theory is that not all infinite sets have the same size. In particular, Cantor showed that the set of real numbers, which can be written as infinite decimals, must be strictly larger than the set of integers.

The set of real numbers, in turn, is smaller than even larger infinities, and so on. To measure the size of infinite sets, Cantor introduced so-called transfinite numbers.

The ever-increasing series of transfinite numbers is denoted by Aleph, the first letter of the Hebrew alphabet, whose mystic nature has been explored by philosophers, theologians and poets alike.

Set theory and Pope Leo XIII

For Cantor, a devout Lutheran Christian, the motivation and justification of his theory of absolute infinities was directly inspired by religion. In fact, he was convinced that the transfinite numbers were communicated to him by God. Moreover, Cantor was deeply concerned about the consequences of his theory for Catholic theology.

Pope Leo XIII, Cantor’s contemporary, encouraged theologians to engage with modern science, to show that the conclusions of science were compatible with religious doctrine. In his extensive correspondence with Catholic theologians, Cantor went to great lengths to argue that his theory does not challenge the status of God as the unique actual infinite being.

On the contrary, he understood his transfinite numbers as increasing the extent of God’s nature, as a “pathway to the throne of God”. Cantor even addressed a letter and several notes on this topic to Leo XIII himself.

Pope Leo XIII. Wikipedia/Braun et Compagnie

For Cantor, absolute infinities lie at the intersection of mathematics and theology. It is striking to consider that one of the most fundamental revolutions in the history of mathematics, the introduction of absolute infinities, was so deeply entangled with religious concerns.

Pope Leo XIV has been explicit that Leo XIII was his inspiration for his choice of pontifical name. Perhaps among an infinite number of potential reasons for the choice, this mathematical link was one.

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

*Credit for article given to Balthasar Grabmayr*

Photo by Markus Krisetya via Unsplash

Statistics professor Johan Ferreira was feeling overwhelmed by the amount of “screen time” involved in online learning in 2021. He imagined students must be feeling the same way, and wondered what he could do to inspire them and make his subject matter more appealing.

One of the topics in statistics is time series analysis: statistical methods to understand trend behaviour in data which is measured over time. There are lots of examples in daily life, from rainfall records to changes in commodity prices, import or exports, or temperature.

Ferreira asked his students to write a short, fictional “bedtime” story using “characters” from time series analysis. The results were collected into a book that is freely available. He tells us more about it.

Why use storytelling to learn about statistics?

I’m fortunate to be something of a creative myself, being a professional oboe player with the Johannesburg Philharmonic Orchestra. It’s a valuable outlet for self-expression. I reflected on what other activity could inspire creativity without compromising the essence of statistical thinking that was required in this particular course I was teaching.

Example of a time series, the kind of data analysed using statistical methods. Author provided (no reuse)

I invited my third-year science and commerce students at the University of Pretoria to take part in a voluntary storytelling exercise, using key concepts in time series analysis as characters. Students got some guidelines but were free to be creative. My colleague and co-editor, Dr Seite Makgai, and I then read, commented on and edited the stories and put them together into an anthology.

Students gave their consent that their stories could be used for research purposes and might be published. Out of a class of over 200 students, over 30 contributions were received; 23 students permitted their work to be included in this volume.

We curated submissions into two sections (Part I: Fables and Fairy Tales and Part II: Fantasy and Sci-Fi) based on the general style and gist of the work.

The project aimed to develop a new teaching resource, inspire students to take ownership of their learning in a creative way, and support them through informal, project-based peer learning.

This collection is written by students, for students. They used personal and cultural contexts relevant to their background and environment to create content that has a solid background in their direct academic interests. And the stories are available without a paywall!

What are some of the characters and stories?

Student Lebogang Malebati wrote Stationaryville and the Two Brothers, a tale about AR(1) and AR(2). In statistics, AR refers to processes in which numerical values are based on past values. The brothers “were both born with special powers, powers that could make them stationary…” and could trick an evil wizard.

David Dodkins wrote Zt and the Shadow-spawn. In this story, Zt (common notation in time series analysis) has a magic amulet that reveals his character growth through a sequence of models and shows the hero’s victory in the face of adversity. He is a function of those that came before him (through an AR process).

Then there’s Nelis Daniels’ story about a shepherd plagued by a wolf called Arma (autoregressive moving average) which kept making sheep disappear.

And Dikelede Rose Motseleng’s modern fable about the love-hate relationship between AR(1) (“more of a linear guy” with a bad habit of predicting the future based on the past) and MA(1), “the type of girl who would always provide you with stationarity (stability).”

What was the impact of the project?

It was a deeply enriching experience for us to see how students see statistics in a context beyond that of the classroom, especially in cases where students reformulated their stories within their own cultural identities or niche interests.

Three particular main impacts stand out for us:

• students have a new additional reference and learning resource for the course content
• new students can refer to the experiences and contextualisation of this content of former students, leading to informal peer learning
• students engage in a cognitive skill (higher-order and creative thinking) that is not frequently considered and included in this field and at this level.

In 2024, shortly after the book was published, we asked students in the time series analysis course of that year to read any one of four stories (related to concepts that were already covered in the course material at that point in time). We asked them to complete a short and informal survey to gauge their experience and insights regarding the potential of this book as a learning resource for them.

The 53 responses we got indicated that most students saw the book as a useful contribution to their learning experience in time series analysis.

Student perceptions of value of stories. Author supplied, Author provided (no reuse)

One positive comment from a student was:

I will always remember that the Random Walk is indeed not stationary but White Noise is. I already knew it, but now I won’t forget it.

Will you build on this in future?

It is definitely valuable to consider similar projects in other branches of statistics, but also, in other disciplines entirely, to develop content by students, for students.

At this stage, we’re having the stories and book translated into languages beyond English. In large classes that are essential to data science (such as statistics and mathematics), many different home languages may be spoken. Students often have to learn in their second, third, or even fourth language. So, this project is proving valuable in making advanced statistical concepts tactile and “at home” via translations.

Our publisher recently let us know that the Setswana translation is complete, with the Sepedi and Afrikaans translations following soon. To our knowledge, it’ll be the first such project not only in the discipline of statistics, but in four of the official languages in South Africa.

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

*Credit for article given to Johan Ferreira*

Rachel Moore

From hand-sized lantern sharks that glow in the deep sea to bus-sized whale sharks gliding through tropical waters, sharks come in all shapes and sizes.

Despite these differences, they all face the same fundamental challenge: how to get oxygen, heat and nutrients to every part of their bodies efficiently.

Our new study, published today in Royal Society Open Science, shows that sharks follow a centuries-old mathematical rule – the two-thirds scaling law – that predicts how body shape changes with size. This tells us something profound about how evolution works – and why size really does matter.

What is the two-thirds scaling law?

The basic idea is mathematical: surface area increases with the square of body length, while volume increases with the cube. That means surface area increases more slowly than volume, and the ratio between the two – crucial for many biological functions – decreases with size.

This matters because many essential life processes happen at the surface: gas exchange in the lungs or gills, such as to take in oxygen or release carbon dioxide, but also heat loss through skin and nutrient uptake in the gut.

These processes depend on surface area, while the demands they must meet – such as the crucial task of keeping the body supplied with oxygen – depend on volume. So, the surface area-to-volume ratio shapes how animals function.

Whale sharks are as big as buses, while dwarf lanternsharks (pictured here) are as small as a human hand. Chip Clark/Smithsonian Institution

Despite its central role in biology, this rule has only ever been rigorously tested in cells, tissues and small organisms such as insects.

Until now.

Why sharks?

Sharks might seem like an unlikely group for testing an old mathematical theory, but they’re actually ideal.

For starters, they span a huge range of sizes, from the tiny dwarf lantern shark (about 20 centimetres long) to the whale shark (which can exceed 20 metres). They also have diverse shapes and lifestyles – hammerheads, reef-dwellers, deep-sea hunters – each posing different challenges for physiology and movement.

Plus, sharks are charismatic, ecologically important and increasingly under threat. Understanding their biology is both scientifically valuable and important for conservation.

Sharks are ecologically important but are increasingly under threat. Rachel Moore

How did we test the rule?

We used high-resolution 3D models to digitally measure surface area and volume in 54 species of sharks. These models were created using open-source CT scans and photogrammetry, which involves using photographs to approximate a 3D structure. Until recently, these techniques were the domain of video game designers and special effects artists, not biologists.

We refined the models in Blender, a powerful 3D software tool, and extracted surface and volume data for each species.

Then we applied phylogenetic regression – a statistical method that accounts for shared evolutionary history – to see how closely shark shapes follow the predictions of the two-thirds rule.

Sharks follow the two-thirds scaling rule almost perfectly, as seen in this 3D representation. Joel Gayford et al

What did we find?

The results were striking: sharks follow the two-thirds scaling rule almost perfectly, with surface area scaling to body volume raised to the power of 0.64 – just a 3% difference from the theoretical 0.67.

This suggests something deeper is going on. Despite their wide range of forms and habitats, sharks seem to converge on the same basic body plan when it comes to surface area and volume. Why?

One explanation is that what are known as “developmental constraints” – limits imposed by how animals grow and form in early life – make it difficult, or too costly, for sharks to deviate from this fundamental pattern.

Changing surface area-to-volume ratios might require rewiring how tissues are allocated during embryonic development, something that evolution appears to avoid unless absolutely necessary.

The scale of sharks

A study of 54 species of sharks shows the ratio of the surface areas and volumes of their bodies follows a mathematical rule called the two-thirds scaling law.

But why does it matter?

This isn’t just academic. Many equations in biology, physiology and climate science rely on assumptions about surface area-to-volume ratios.

These equations are used to model how animals regulate temperature, use oxygen, and respond to environmental stress. Until now, we haven’t had accurate data from large animals to test those assumptions. Our findings give researchers more confidence in using these models – not just for sharks, but potentially for other groups too.

As we face accelerating climate change and biodiversity loss, understanding how animals of all sizes interact with their environments has never been more urgent.

This study, powered by modern imaging tech and some old-school curiosity, brings us one step closer to that goal.

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

*Credit for article given to Jodie L. Rummer & Joel Gayford*

A painless, non-invasive brain stimulation technique can significantly improve how young adults learn maths, my colleagues and I found in a recent study. In a paper in PLOS Biology, we describe how this might be most helpful for those who are likely to struggle with mathematical learning because of how their brain areas involved in this skill communicate with each other.

Maths is essential for many jobs, especially in science, technology, engineering and finance. However, a 2016 OECD report suggested that a large proportion of adults in developed countries (24% to 29%) have maths skills no better than a typical seven-year-old. This lack of numeracy can contribute to lower income, poor health, reduced political participation and even diminished trust in others.

Education often widens rather than closes the gap between high and low achievers, a phenomenon known as the Matthew effect. Those who start with an advantage, such as being able to read more words when starting school, tend to pull further ahead. Stronger educational achievement has been also associated with socioeconomic status, higher motivation and greater engagement with material learned during a class.

Biological factors, such as genes, brain connectivity, and chemical signalling, have been shown in some studies to play a stronger role in learning outcomes than environmental ones. This has been well-documented in different areas, including maths, where differences in biology may explain educational achievements.

To explore this question, we recruited 72 young adults (18–30 years old) and taught them new maths calculation techniques over five days. Some received a placebo treatment. Others received transcranial random noise stimulation (tRNS), which delivers gentle electrical currents to the brain. It is painless and often imperceptible, unless you focus hard to try and sense it.

It is possible tRNS may cause long term side effects, but in previous studies my team assessed participants for cognitive side effects and found no evidence for it.

Participants who received tRNS were randomly assigned to receive it in one of two different brain areas. Some received it over the dorsolateral prefrontal cortex, a region critical for memory, attention, or when we acquire a new cognitive skill. Others had tRNS over the posterior parietal cortex, which processes maths information, mainly when the learning has been accomplished.

Before and after the training, we also scanned their brains and measured levels of key neurochemicals such as gamma-aminobutyric acid (gaba), which we showed previously, in a 2021 study, to play a role in brain plasticity and learning, including maths.

Some participants started with weaker connections between the prefrontal and parietal brain regions, a biological profile that is associated with poorer learning. The study results showed these participants made significant gains in learning when they received tRNS over the prefrontal cortex.

Stimulation helped them catch up with peers who had stronger natural connectivity. This finding shows the critical role of the prefrontal cortex in learning and could help reduce educational inequalities that are grounded in neurobiology.

How does this work? One explanation lies in a principle called stochastic resonance. This is when a weak signal becomes clearer when a small amount of random noise is added.

In the brain, tRNS may enhance learning by gently boosting the activity of underperforming neurons, helping them get closer to the point at which they become active and send signals. This is a point known as the “firing threshold”, especially in people whose brain activity is suboptimal for a task like maths learning.

It is important to note what this technique does not do. It does not make the best learners even better. That is what makes this approach promising for bridging gaps, not widening them. This form of brain stimulation helps level the playing field.

Our study focused on healthy, high-performing university students. But in similar studies on children with maths learning disabilities (2017) and with attention-deficit/hyperactivity disorder (2023) my colleagues and I found tRNS seemed to improve their learning and performance in cognitive training.

I argue our findings could open a new direction in education. The biology of the learner matters, and with advances in knowledge and technology, we can develop tools that act on the brain directly, not just work around it. This could give more people the chance to get the best benefit from education.

In time, perhaps personalised, brain-based interventions like tRNS could support learners who are being left behind not because of poor teaching or personal circumstances, but because of natural differences in how their brains work.

Of course, very often education systems aren’t operating to their full potential because of inadequate resources, social disadvantage or systemic barriers. And so any brain-based tools must go hand-in-hand with efforts to tackle these obstacles.

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

*Credit for article given to Roi Cohen Kadosh*

Algebra often involves manipulating numbers or other objects using operations like addition and multiplication. Flavio Coelho/Moment via Getty Images

You scrambled up a Rubik’s cube, and now you want to put it back in order. What sequence of moves should you make?

Surprise: You can answer this question with modern algebra.

Most folks who have been through high school mathematics courses will have taken a class called algebra – maybe even a sequence of classes called algebra I and algebra II that asked you to solve for x. The word “algebra” may evoke memories of complicated-looking polynomial equations like ax² + bx + c = 0 or plots of polynomial functions like y = ax² + bx + c.

You might remember learning about the quadratic formula to figure out the solutions to these equations and find where the plot crosses the x-axis, too.

Graph of a quadratic equation and its roots via the quadratic formula. Jacob Rus, CC BY-SA

Equations and plots like these are part of algebra, but they’re not the whole story. What unifies algebra is the practice of studying things – like the moves you can make on a Rubik’s cube or the numbers on a clock face you use to tell time – and the way they behave when you put them together in different ways. What happens when you string together the Rubik’s cube moves or add up numbers on a clock?

In my work as a mathematician, I’ve learned that many algebra questions come down to classifying objects by their similarities.

Sets and groups

How did equations like ax² + bx + c = 0 and their solutions lead to abstract algebra?

The short version of the story is that mathematicians found formulas that looked a lot like the quadratic formula for polynomial equations where the highest power of x was three or four. But they couldn’t do it for five. It took mathematician Évariste Galois and techniques he developed – now called group theory – to make a convincing argument that no such formula could exist for polynomials with a highest power of five or more.

So what is a group, anyway?

It starts with a set, which is a collection of things. The fruit bowl in my kitchen is a set, and the collection of things in it are pieces of fruit. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 also form a set. Sets on their own don’t have too many properties – that is, characteristics – but if we start doing things to the numbers 1 through 12, or the fruit in the fruit bowl, it gets more interesting.

In clock addition, 3 + 12 = 3. OpenStax, CC BY-SA

Let’s call this set of numbers 1 through 12 “clock numbers.” Then, we can define an addition function for the clock numbers using the way we tell time. That is, to say “3 + 11 = 2” is the way we would add 3 and 11. It feels weird, but if you think about it, 11 hours past 3 o’clock is 2 o’clock.

Clock addition has some nice properties. It satisfies:

closure, where adding things in the set gives you something else in the set,

identity, where there’s an element that doesn’t change the value of other elements in the set when added – adding 12 to any number will equal that same number,

associativity, where you can add wherever you want in the set,

inverses, where you can undo whatever an element does, and

commutativity, where you can change the order of which clock numbers you add up without changing the outcome: a + b = b + a.

By satisfying all these properties, mathematicians can consider clock numbers with clock addition a group. In short, a group is a set with some way of combining the elements layered on top. The set of fruit in my fruit bowl probably can’t be made into a group easily – what’s a banana plus an apple? But we can make a set of clock numbers into a group by showing that clock addition is a way of taking two clock numbers and getting to a new one that satisfies the rules outlined above.

Rings and fields

Along with groups, the two other fundamental types of algebraic objects you would study in an introduction to modern algebra are rings and fields.

We could introduce a second operation for the clock numbers: clock multiplication, where 2 times 7 is 2, because 14 o’clock is the same as 2 o’clock. With clock addition and clock multiplication, the clock numbers meet the criteria for what mathematicians call a ring. This is primarily because clock multiplication and clock addition together satisfy a key component that defines a ring: the distributive property, where a(b + c) = ab + ac. Lastly, fields are rings that satisfy even more conditions.

At the turn of the 20th century, mathematicians David Hilbert and Emmy Noether – who were interested in understanding how the principles in Einstein’s relativity worked mathematically – unified algebra and showed the utility of studying groups, rings and fields.

It’s all fun and games until you do the math

Groups, rings and fields are abstract, but they have many useful applications.

For example, the symmetries of molecular structures are categorized by different point groups. A point group describes ways to move a molecule in space so that even if you move the individual atoms, the end result is indistinguishable from the molecule you started with.

The water molecule H₂O can be flipped horizontally and the end result is indistinguishable from the original position. Courtney Gibbons, CC BY-SA

But let’s take a different example that uses rings instead of groups. You can set up a pretty complicated set of equations to describe a Sudoku puzzle: You need 81 variables to represent each place you can put a number in the grid, polynomial expressions to encode the rules of the game, and polynomial expressions that take into account the clues already on the board.

To get the spaces on the game board and the 81 variables to correspond nicely, you can use two subscripts to associate the variable with a specific place on the board, like using x₃₅ to represent the cell in the third row and fifth column.

The first entry must be one of the numbers 1 through 9, and we represent that relationship with (x₁₁ – 1)(x₁₁ – 2)(x₁₁ – 3) ⋅⋅⋅ (x₁₁ – 9). This expression is equal to zero if and only if you followed the rules of the game. Since every space on the board follows this rule, that’s already 81 equations just to say, “Don’t plug in anything other than 1 through 9.”

The rule “1 through 9 each appear exactly once in the top row” can be captured with some sneaky pieces of algebraic thinking. The sum of the top row is going to add up to 45, which is to say x₁₁ + x₁₂ + ⋅⋅⋅ + x₁₉ – 45 will be zero, and the product of the top row is going to be the product of 1 through 9, which is to say x₁₁ x₁₂ ⋅⋅⋅ x₁₉ – 9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 will be zero.

If you’re thinking that it takes more time to set up all these rules than it does to solve the puzzle, you’re not wrong.

Turning Sudoku into algebra takes a fair bit of work. Courtney Gibbons

What do we get by doing this complicated translation into algebra? Well, we get to use late-20th century algorithms to figure out what numbers you can plug into the board that satisfy all the rules and all the clues. These algorithms are based on describing the structure of the special ring – called an ideal – these game board clues make within the larger ring. The algorithms will tell you if there’s no solution to the puzzle. If there are multiple solutions, the algorithms will find them all.

This is a small example where setting up the algebra is harder than just doing the puzzle. But the techniques generalize widely. You can use algebra to tackle problems in artificial intelligence, robotics, cryptography, quantum computing and so much more – all with the same bag of tricks you’d use to solve the Sudoku puzzle or Rubik’s cube.

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

*Credit for article given to Courtney Gibbons*

Prime numbers are numbers that are not products of smaller whole numbers. Jeremiah Bartz

A shard of smooth bone etched with irregular marks dating back 20,000 years puzzled archaeologists until they noticed something unique – the etchings, lines like tally marks, may have represented prime numbers. Similarly, a clay tablet from 1800 B.C.E. inscribed with Babylonian numbers describes a number system built on prime numbers.

As the Ishango bone, the Plimpton 322 tablet and other artifacts throughout history display, prime numbers have fascinated and captivated people throughout history. Today, prime numbers and their properties are studied in number theory, a branch of mathematics and active area of research today.

A history of prime numbers

Some scientists guess that the markings on the Ishango bone represent prime numbers. Joeykentin/Wikimedia Commons, CC BY-SA

Informally, a positive counting number larger than one is prime if that number of dots can be arranged only into a rectangular array with one column or one row. For example, 11 is a prime number since 11 dots form only rectangular arrays of sizes 1 by 11 and 11 by 1. Conversely, 12 is not prime since you can use 12 dots to make an array of 3 by 4 dots, with multiple rows and multiple columns. Math textbooks define a prime number as a whole number greater than one whose only positive divisors are only 1 and itself.

Math historian Peter S. Rudman suggests that Greek mathematicians were likely the first to understand the concept of prime numbers, around 500 B.C.E.

Around 300 B.C.E., the Greek mathematician and logician Euclid proved that there are infinitely many prime numbers. Euclid began by assuming that there is a finite number of primes. Then he came up with a prime that was not on the original list to create a contradiction. Since a fundamental principle of mathematics is being logically consistent with no contradictions, Euclid then concluded that his original assumption must be false. So, there are infinitely many primes.

The argument established the existence of infinitely many primes, however it was not particularly constructive. Euclid had no efficient method to list all the primes in an ascending list.

In the middle ages, Arab mathematicians advanced the Greeks’ theory of prime numbers, referred to as hasam numbers during this time. The Persian mathematician Kamal al-Din al-Farisi formulated the fundamental theorem of arithmetic, which states that any positive integer larger than one can be expressed uniquely as a product of primes.

From this view, prime numbers are the basic building blocks for constructing any positive whole number using multiplication – akin to atoms combining to make molecules in chemistry.

Prime numbers can be sorted into different types. In 1202, Leonardo Fibonacci introduced in his book “Liber Abaci: Book of Calculation” prime numbers of the form (2p – 1) where p is also prime.

Prime numbers, when expressed as that number of dots, can be arranged only in a single row or column, rather than a square or rectangle. David Eppstein/Wikimedia Commons

Today, primes in this form are called Mersenne primes after the French monk Marin Mersenne. Many of the largest known primes follow this format.

Several early mathematicians believed that a number of the form (2p – 1) is prime whenever p is prime. But in 1536, mathematician Hudalricus Regius noticed that 11 is prime but not (211 – 1), which equals 2047. The number 2047 can be expressed as 23 times 89, disproving the conjecture.

While not always true, number theorists realized that the (2p – 1) shortcut often produces primes and gives a systematic way to search for large primes.

The search for large primes

The number (2p – 1) is much larger relative to the value of p and provides opportunities to identify large primes.

When the number (2p – 1) becomes sufficiently large, it is much harder to check whether (2p – 1) is prime – that is, if (2p – 1) dots can be arranged only into a rectangular array with one column or one row.

Fortunately, Édouard Lucas developed a prime number test in 1878, later proved by Derrick Henry Lehmer in 1930. Their work resulted in an efficient algorithm for evaluating potential Mersenne primes. Using this algorithm with hand computations on paper, Lucas showed in 1876 that the 39-digit number (2127 – 1) equals 170,141,183,460,469,231,731,687,303,715,884,105,727, and that value is prime

Also known as M127, this number remains the largest prime verified by hand computations. It held the record for largest known prime for 75 years.

Researchers began using computers in the 1950s, and the pace of discovering new large primes increased. In 1952, Raphael M. Robinson identified five new Mersenne primes using a Standard Western Automatic Computer to carry out the Lucas-Lehmer prime number tests.

As computers improved, the list of Mersenne primes grew, especially with the Cray supercomputer’s arrival in 1964. Although there are infinitely many primes, researchers are unsure how many fit the type (2p – 1) and are Mersenne primes.

By the early 1980s, researchers had accumulated enough data to confidently believe that infinitely many Mersenne primes exist. They could even guess how often these prime numbers appear, on average. Mathematicians have not found proof so far, but new data continues to support these guesses.

George Woltman, a computer scientist, founded the Great Internet Mersenne Prime Search, or GIMPS, in 1996. Through this collaborative program, anyone can download freely available software from the GIMPS website to search for Mersenne prime numbers on their personal computers. The website contains specific instructions on how to participate.

GIMPS has now identified 18 Mersenne primes, primarily on personal computers using Intel chips. The program averages a new discovery about every one to two years.

The largest known prime

Luke Durant, a retired programmer, discovered the current record for the largest known prime, (2136,279,841 – 1), in October 2024

Referred to as M136279841, this 41,024,320-digit number was the 52nd Mersenne prime identified and was found by running GIMPS on a publicly available cloud-based computing network.

This network used Nvidia chips and ran across 17 countries and 24 data centers. These advanced chips provide faster computing by handling thousands of calculations simultaneously. The result is shorter run times for algorithms such as prime number testing.

New and increasingly powerful computer chips have allowed prime-number hunters to find increasingly larger primes. Fritzchens Fritz/Flickr

The Electronic Frontier Foundation is a civil liberty group that offers cash prizes for identifying large primes. It awarded prizes in 2000 and 2009 for the first verified 1 million-digit and 10 million-digit prime numbers.

Large prime number enthusiasts’ next two challenges are to identify the first 100 million-digit and 1 billion-digit primes. EFF prizes of US$150,000 and $250,000, respectively, await the first successful individual or group.

Eight of the 10 largest known prime numbers are Mersenne primes, so GIMPS and cloud computing are poised to play a prominent role in the search for record-breaking large prime numbers.

Large prime numbers have a vital role in many encryption methods in cybersecurity, so every internet user stands to benefit from the search for large prime numbers. These searches help keep digital communications and sensitive information safe.

This story was updated on May 30, 2025 to correct the name of the Greek mathematician Euclid and to correct the factors of 2047.

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*Credit for article given to Jeremiah Bartz*

School mathematics in South Africa is often seen as a sign of the health of the education system more generally. Under the racial laws of apartheid, until 1994, African people were severely restricted from learning maths. Tracking the changes in maths performance is a measure of how far the country has travelled in overcoming past injustices. Maths is also an essential foundation for meeting the challenges of the future, like artificial intelligence, climate change, energy and sustainable development.

Here, education researcher Vijay Reddy takes stock of South Africa’s mathematical capabilities. She reports on South African maths performance at grades 5 (primary school) and 9 (secondary school) in the Trends in International Mathematics and Science Study (TIMSS) and examines the gender gaps in mathematics achievement

What was unusual about the latest TIMSS study?

The study is conducted every four years. South Africa has participated in it at the secondary phase since 1995 and at the primary phase since 2015. The period between the 2019 and 2023 cycles was characterised by the onset of the COVID-19 pandemic, social distancing and school closures.

The Department of Basic Education estimated that an average of 152 school contact days were lost in 2020 and 2021. South Africa was among the countries with the highest school closures, along with Colombia, Costa Rica and Brazil. At the other end, European countries lost fewer than 50 days.

Some academics measured the extent of learning losses for 2020 and 2021 school closures, but there were no models to estimate subsequent learning losses. We can get some clues of the effects on learning over four years, by comparing patterns within South Africa against the other countries.

How did South African learners (and others) perform in the maths study?

The South African grade 9 mathematics achievement improved by 8 points from 389 in TIMSS 2019 to 397 in 2023. From the trends to TIMSS 2019, we had predicted a mathematics score of 403 in 2023.

For the 33 countries that participated in both the 2019 and 2023 secondary school TIMSS cycles, the average achievement decreased by 9 points from 491 in 2019 to 482 to 2023. Only three countries showed significant increases (United Arab Emirates, Romania and Sweden). There were no significant changes in 16 countries (including South Africa). There were significant decreases in 14 countries.

Based on these numbers, it would seem, on the face of it at least, that South Africa weathered the COVID-19 losses better than half the other countries.

However, the primary school result patterns were different. For South African children, there was a significant drop in mathematics achievement by 12 points, from 374 in 2019 to 362 in 2023. As expected, the highest decreases were in the poorer, no-fee schools.

Of the 51 countries that participated in both TIMSS 2019 and 2023, the average mathematics achievement score over the two cycles was similar. There were no significant achievement changes in 22 countries, a significant increase in 15 countries, and a significant decrease in 14 countries (including South Africa).

So, it seems that South African primary school learners suffered adverse learning effects over the two cycles.

The increase in achievement in secondary school and decrease in primary school was unexpected. These reasons for the results may be that secondary school learners experienced more school support compared with primary schools, or were more mature and resilient, enabling them to recover from the learning losses experienced during COVID-19. Learners in primary schools, especially poorer schools, may have been more affected by the loss of school contact time and had less support to fully recover during this time.

This pattern may also be due to poor reading and language skills as well as lack of familiarity with this type of test.

Does gender make a difference?

There is an extant literature indicating that globally boys are more likely to outperform girls in maths performance.

But in South African primary schools, girls outscore boys in both mathematics and reading. Girls significantly outscored boys by an average of 29 points for mathematics (TIMSS) and by 49 points for reading in the 2021 Progress in International Reading Study, PIRLS.

These patterns need further exploration. Of the 58 countries participating in TIMSS at primary schools, boys significantly outscored girls in 40 countries, and there were no achievement differences in 17 countries. South Africa was the only country where the girls significantly outscored boys. In Kenya, Zimbabwe, Zambia and Mozambique, the Southern and Eastern Africa Consortium for Monitoring Educational Quality (SEACMEQ) reading scores are similar for girls and boys, while the boys outscore girls in mathematics. In Botswana, girls outscore boys in reading and mathematics, but the gender difference is much smaller.

In secondary schools, girls continue to outscore boys, but the gap drops to 8 points. Of the 42 TIMSS countries, boys significantly outscored girls in maths in 21 countries; there were no significant difference in 17 countries; and girls significantly outscored boys in only four countries (South Africa, Palestine, Oman, Bahrain).

In summary, the South African primary school achievement trend relative to secondary school is unexpected and requires further investigation. It seems that as South African learners get older, they acquire better skills in how to learn, read and take tests to achieve better results. Results from lower grades should be used cautiously to predict subsequent educational outcomes.

Unusually, in primary schools, there is a big gender difference for mathematics achievement favouring girls. The gender difference persists to grade 9, but the extent of the difference decreases. As learners, especially boys, progress through their education system they seem to make up their learning shortcomings and catch up.

The national mathematics picture would look much better if boys and girls performed at the same level from primary school, suggesting the importance of interventions in primary schools, especially focusing on boys.

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

*Credit for article given to Vijay Reddy*

You can describe the shape you live on in multiple dimensions. vkulieva/iStock via Getty Images Plus

When you look at your surrounding environment, it might seem like you’re living on a flat plane. After all, this is why you can navigate a new city using a map: a flat piece of paper that represents all the places around you. This is likely why some people in the past believed the earth to be flat. But most people now know that is far from the truth.

You live on the surface of a giant sphere, like a beach ball the size of the Earth with a few bumps added. The surface of the sphere and the plane are two possible 2D spaces, meaning you can walk in two directions: north and south or east and west.

What other possible spaces might you be living on? That is, what other spaces around you are 2D? For example, the surface of a giant doughnut is another 2D space.

Through a field called geometric topology, mathematicians like me study all possible spaces in all dimensions. Whether trying to design secure sensor networks, mine data or use origami to deploy satellites, the underlying language and ideas are likely to be that of topology.

The shape of the universe

When you look around the universe you live in, it looks like a 3D space, just like the surface of the Earth looks like a 2D space. However, just like the Earth, if you were to look at the universe as a whole, it could be a more complicated space, like a giant 3D version of the 2D beach ball surface or something even more exotic than that.

A doughnut, also called a torus, is a shape that you can move across in two directions, just like the surface of the Earth. YassineMrabet via Wikimedia Commons, CC BY-NC-SA

While you don’t need topology to determine that you are living on something like a giant beach ball, knowing all the possible 2D spaces can be useful. Over a century ago, mathematicians figured out all the possible 2D spaces and many of their properties.

In the past several decades, mathematicians have learned a lot about all of the possible 3D spaces. While we do not have a complete understanding like we do for 2D spaces, we do know a lot. With this knowledge, physicists and astronomers can try to determine what 3D space people actually live in.

While the answer is not completely known, there are many intriguing and surprising possibilities. The options become even more complicated if you consider time as a dimension.

To see how this might work, note that to describe the location of something in space – say a comet – you need four numbers: three to describe its position and one to describe the time it is in that position. These four numbers are what make up a 4D space.

Now, you can consider what 4D spaces are possible and in which of those spaces do you live.

Topology in higher dimensions

At this point, it may seem like there is no reason to consider spaces that have dimensions larger than four, since that is the highest imaginable dimension that might describe our universe. But a branch of physics called string theory suggests that the universe has many more dimensions than four.

There are also practical applications of thinking about higher dimensional spaces, such as robot motion planning. Suppose you are trying to understand the motion of three robots moving around a factory floor in a warehouse. You can put a grid on the floor and describe the position of each robot by their x and y coordinates on the grid. Since each of the three robots requires two coordinates, you will need six numbers to describe all of the possible positions of the robots. You can interpret the possible positions of the robots as a 6D space.

As the number of robots increases, the dimension of the space increases. Factoring in other useful information, such as the locations of obstacles, makes the space even more complicated. In order to study this problem, you need to study high-dimensional spaces.

There are countless other scientific problems where high-dimensional spaces appear, from modeling the motion of planets and spacecraft to trying to understand the “shape” of large datasets.

Tied up in knots

Another type of problem topologists study is how one space can sit inside another.

For example, if you hold a knotted loop of string, then we have a 1D space (the loop of string) inside a 3D space (your room). Such loops are called mathematical knots.

The study of knots first grew out of physics but has become a central area of topology. They are essential to how scientists understand 3D and 4D spaces and have a delightful and subtle structure that researchers are still trying to understand.

Knots are examples of spaces that sit inside other spaces. Jkasd/Wikimedia Commons

In addition, knots have many applications, ranging from string theory in physics to DNA recombination in biology to chirality in chemistry.

What shape do you live on?

Geometric topology is a beautiful and complex subject, and there are still countless exciting questions to answer about spaces.

For example, the smooth 4D Poincaré conjecture asks what the “simplest” closed 4D space is, and the slice-ribbon conjecture aims to understand how knots in 3D spaces relate to surfaces in 4D spaces.

Topology is currently useful in science and engineering. Unraveling more mysteries of spaces in all dimensions will be invaluable to understanding the world in which we live and solving real-world problems.

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

*Credit for article given to John Etnyre*

Michael Jung/ Shutterstock

There is a persistent gender gap in Australian schools. Boys, on average, outperform girls in maths.

We see this in national tests such as NAPLAN, as well as international assessments.

New Australian Council for Educational Research analysis by my colleague Catherine Underwood shows how boys, on average, are also more confident and positive about maths than girls.

What can parents do to help their children feel more confident about this core subject?

Boys outperform girls in maths

An important measure of students’ maths performance is the OECD’s Programme for International Assessment (PISA) test. Run every three years, it measures 15-year-olds’ ability to apply their maths, science and reading knowledge to real-world situations.

In 2022, 53% of Australian male students achieved the PISA national proficiency standard in maths, compared with 48% of female students. The gender gap on average scores was also greater in Australia than across the OECD.

As part of PISA, students also completed a questionnaire about their attitudes to learning. ACER’s new analysis uses data from the questionnaire to look at Australian students’ confidence in maths and how this differs between girls and boys.

Boys outperformed girls in maths skills in the most recent PISA test. Monkey Business Images/ Shutterstock

Why is confidence so important?

Research suggests students’ confidence has an impact on their academic performance. Researchers can call this “self-efficacy”, or the belief in your ability to successfully perform tasks and solve problems.

Students with high mathematical self-efficacy embrace challenges, use effective problem-solving strategies, and persevere despite difficulties. Those with low self-efficacy may avoid tasks, experience anxiety, and ultimately underperform due to a lack of confidence in their maths abilities.

We can see this in the 2022 PISA results. Girls in the top quarter on the self-rated “self-efficacy index” scored an average of 568 points on the PISA maths performance test, a staggering 147 points higher than the average for girls in the lowest quarter on the index.

For boys, the benefit of confidence was even more pronounced. Those in the top quarter of the index scored 159 points on average higher in maths performance than those in the lowest quarter.

Boys are more confident than girls

The PISA questionnaire asked students how confident they felt about having to do a range of formal and applied maths tasks.

Students showed similar levels of confidence solving formal maths tasks such as equations. But male students, on average, showed they were more confident than female students with applied mathematics tasks such as:

• finding distances using a map
• calculating a power consumption rate
• calculating how much more expensive a computer would be after adding tax
• calculating how many square metres of tiles are needed to cover a floor.

What about attitude?

The PISA data also shows Australian boys, on average, have more positive attitudes towards maths than girls.

For example, in response to the statement “mathematics is easy for me” only 41% of female students agreed, compared with 55% of male students.

In response to “mathematics is one of my favourite subjects”, 37% of female students agreed, compared with 49% of males.

But in response to “I want to do well in my mathematics class”, 91% of female students agreed, compared to 92% of males.

What can parents do at home to help?

It is troubling that girls, on average, show consistently lower levels of confidence about maths tasks.

This comes on top of other PISA questionnaire results that have shown in general (not just around maths) that a higher proportion of girls than boys say they feel nervous approaching exams.

We want all students to have a positive relationship with maths, where they can appreciate maths skills are important in many aspects of their lives, and they’re willing to have a go to develop them.

Recently, we collaborated with the Victorian Academy of Teaching and Leadership on resources for teachers, students and parents that focus on addressing maths anxiety.

Research shows how we talk about maths at home is important in shaping students’ attitudes and persistence. Parents can help create a positive atmosphere around maths by:

• dispelling “maths myths”, such as the idea maths ability is fixed and no amount of effort or practise can improve it
• talking about how making mistakes is a normal part of learning
• thinking about about how we forgive mistakes in other areas (such as sport, art or science): how can we treat maths mistakes in a similar way?
• telling your child they have done a good job when they put effort into their maths learning.

Parents can also help their children even if they don’t know the answers to maths problems. It’s perfectly fine to say, “I’m not sure how to do that one but who can we ask for help? Let’s talk to the teacher.”

Modelling a “help-seeking” approach lets children know that it’s OK not to know the answer, the key is to persist and try.

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

*Credit for article given to Sarah Buckley*

Edward Lorenz’s mathematical weather model showed solutions with a butterfly-like shape. Wikimol

In 1972, the US meteorologist Edward Lorenz asked a now-famous question:

Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

Over the next 50 years, the so-called “butterfly effect” captivated the public imagination. It has appeared in movies, books, motivational and inspirational speeches, and even casual conversation.

The image of the tiny flapping butterfly has come to stand for the outsized impact of small actions, or even the inherent unpredictability of life itself. But what was Lorenz – who is now remembered as the founder of the branch of mathematics called chaos theory – really getting at?

A simulation goes wrong

Our story begins in the 1960s, when Lorenz was trying to use early computers to predict the weather. He had built a basic weather simulation that used a simplified model, designed to calculate future weather patterns.

One day, while re-running a simulation, Lorenz decided to save time by restarting the calculations from partway through. He manually inputted the numbers from halfway through a previous printout.

But instead of inputting, let’s say, 0.506127, he entered 0.506 as the starting point of the calculations. He thought the small difference would be insignificant.

He was wrong. As he later told the story:

I started the computer again and went out for a cup of coffee. When I returned about an hour later, after the computer had generated about two months of data, I found that the new solution did not agree with the original one. […] I realized that if the real atmosphere behaved in the same manner as the model, long-range weather prediction would be impossible, since most real weather elements were certainly not measured accurately to three decimal places.

There was no randomness in Lorenz’s equations. The different outcome was caused by the tiny change in the input numbers.

Lorenz realised his weather model – and by extension, the real atmosphere – was extremely sensitive to initial conditions. Even the smallest difference at the start – even something as small as the flap of a butterfly’s wings – could amplify over time and make accurate long-term predictions impossible.

The ‘Lorenz Attractor’ found in models of a chaotic weather system has a characteristic butterfly shape. Milad Haghani, CC BY

Lorenz initially used “the flap of a seagull’s wings” to describe his findings, but switched to “butterfly” after noticing a remarkable feature of the solutions to his equations.

In his weather model, when he plotted the solutions, they formed a swirling, three-dimensional shape that never repeated itself. This shape — called the Lorenz attractor — looked strikingly like a butterfly with two looping wings.

Welcome to chaos

Lorenz’s efforts to understand weather led him to develop chaos theory, which deals with systems that follow fixed rules but behave in ways that seem unpredictable.

These systems are deterministic, which means the outcome is entirely governed by initial conditions. If you know the starting point and the rules of the system, you should be able to predict the future outcome.

There is no randomness involved. For example, a pendulum swinging back and forth is deterministic — it operates based on the laws of physics.

Systems governed by the laws of nature, where human actions don’t play a central role, are often deterministic. In contrast, systems involving humans, such as financial markets, are not typically considered deterministic due to the unpredictable nature of human behaviour.

A chaotic system is a system that is deterministic but nevertheless behaves unpredictably. The unpredictability happens because chaotic systems are extremely sensitive to initial conditions. Even the tiniest differences at the start can grow over time and lead to wildly different outcomes

Chaos is not the same as randomness. In a random system, outcomes have no definitive underlying order. In a chaotic system, however, there is order, but it’s so complex it appears disordered.

A misunderstood meme

Like many scientific ideas in popular culture, the butterfly effect has often been misunderstood and oversimplified.

One common misconception is that the butterfly effect implies every small action leads to massive consequences. In reality, not all systems are chaotic, and for systems that aren’t, small changes usually result in small effects.

Another is that the butterfly effect carries a sense of inevitability, as though every butterfly in the Amazon is triggering tornadoes in Texas with each flap of its wings.

This is not at all correct. It’s simply a metaphor pointing out that small changes in chaotic systems can amplify over time, making long-term outcomes impossible to predict with precision.

Taming butterflies

Systems that are very sensitive to initial conditions are very hard to predict. Weather systems are still tricky, for example

Forecasts have improved a lot since Lorenz’s early efforts, but they are still only reliable for a week or so. After that, small errors or imprecisions in the starting data grow larger and larger, eventually making the forecast inaccurate.

To deal with the butterfly effect, meteorologists use a method called ensemble forecasting. They run many simulations, each starting with slightly different initial conditions.

By comparing the results, they can estimate the range of possible outcomes and their likelihoods. For example, if most simulations predict rain but a few predict sunshine, forecasters can report a high probability of rain.

However, even this approach works only up to a point. As time goes on, the predictions from the models diverge rapidly. Eventually, the differences between the simulations become so large that even their average no longer provides useful information about what will happen on a given day at a given location.

A butterfly effect for the butterfly effect?

The journey of the butterfly effect from a rigorous scientific concept to a widely popular metaphor highlights how ideas can evolve as they move beyond their academic roots.

While this has helped bring attention to a complex scientific concept, it has also led to oversimplifications and misconceptions about what it really means.

Attaching a metaphor to a scientific phenomenon and releasing it into popular culture can lead to its gradual distortion.

Any tiny inaccuracies or imprecision in the initial description can be amplified over time, until the final outcome is a long way from reality. Sound familiar?

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

*Credit for article given to Milad Haghani*