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Prime numbers, the building blocks of mathematics, have fascinated for centuries − now technology is revolutionizing the search for them

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theconversation.com – Jeremiah Bartz, Associate Professor of Mathematics, University of North Dakota – 2025-05-30 07:47:00


Prime numbers, numbers greater than one divisible only by one and themselves, have fascinated humanity for millennia, evidenced by artifacts like the 20,000-year-old Ishango bone and the Babylonian Plimpton 322 tablet. Greek mathematicians around 500 B.C.E. first understood primes, while Euler proved their infinitude circa 300 B.C.E. Arab scholars advanced prime theory, including the fundamental theorem of arithmetic. Mersenne primes, of form (2^p – 1), offer a key to finding large primes. The Lucas-Lehmer test enables efficient identification, enhanced by computers since the 1950s. Collaborative efforts like GIMPS have discovered many large primes, with the current largest prime found in 2024, critical for encryption and cybersecurity.

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

Jeremiah Bartz, University of North Dakota

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

A long, thin shard of bone with small lines scratched into it.
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.

a diagram showing prime numbers as dots in rows, with composite numbers as dots arranged in rectangles of at least two rows of dots, with the same number of dots in each row.
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

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.

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.

A small rectangle metal chip reading 'nVIDIA'
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.The Conversation

Jeremiah Bartz, Associate Professor of Mathematics, University of North Dakota

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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Note: The following A.I. based commentary is not part of the original article, reproduced above, but is offered in the hopes that it will promote greater media literacy and critical thinking, by making any potential bias more visible to the reader –Staff Editor.

Political Bias Rating: Centrist

The article presents a factual, educational overview of the history and significance of prime numbers, focusing on mathematics and technological advancements without promoting any political or ideological stance. Its tone is neutral and informative, aimed at explaining mathematical concepts and recent developments in prime number research. The content does not include partisan language or viewpoints and remains centered on scientific progress and historical context, making it a balanced, non-partisan piece.

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Fears that falling birth rates in US could lead to population collapse are based on faulty assumptions

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theconversation.com – Leslie Root, Assistant Professor of Research, Institute of Behavioral Science, University of Colorado Boulder – 2025-07-25 07:34:00


Pronatalism, the belief that low birth rates must be reversed, is gaining attention amid declining fertility in the U.S. and globally. However, demographers argue population collapse is neither imminent nor inevitable. Total fertility rates fluctuate and do not perfectly predict lifetime childbearing, especially with delayed births and advances in fertility treatments. Although U.S. fertility rates hit a historic low of 1.6 in 2024, the average number of children women have by age 40 remains around two. Population projections foresee growth, aided by immigration. Economic concerns about aging populations overlook factors like rising labor participation among older adults, women’s workforce gains, and the importance of economic policies and technology.

Unfortunately for demographers, birth rates are hard to predict far into the future.
gremlin/E+ via Getty Images

Leslie Root, University of Colorado Boulder; Karen Benjamin Guzzo, University of North Carolina at Chapel Hill, and Shelley Clark, McGill University

Pronatalism – the belief that low birth rates are a problem that must be reversed – is having a moment in the U.S.

As birth rates decline in the U.S. and throughout the world, voices from Silicon Valley to the White House are raising concerns about what they say could be the calamitous effects of steep population decline on the economy. The Trump administration has said it is seeking ideas on how to encourage Americans to have more children as the U.S. experiences its lowest total fertility rate in history, down about 25% since 2007.

As demographers who study fertility, family behaviors and childbearing intentions, we can say with certainty that population decline is not imminent, inevitable or necessarily catastrophic.

The population collapse narrative hinges on three key misunderstandings. First, it misrepresents what standard fertility measures tell us about childbearing and makes unrealistic assumptions that fertility rates will follow predictable patterns far into the future. Second, it overstates the impact of low birth rates on future population growth and size. Third, it ignores the role of economic policies and labor market shifts in assessing the impacts of low birth rates.

Fertility fluctuations

Demographers generally gauge births in a population with a measure called the total fertility rate. The total fertility rate for a given year is an estimate of the average number of children that women would have in their lifetime if they experienced current birth rates throughout their childbearing years.

Fertility rates are not fixed – in fact, they have changed considerably over the past century. In the U.S., the total fertility rate rose from about 2 births per woman in the 1930s to a high of 3.7 births per woman around 1960. The rate then dipped below 2 births per woman in the late 1970s and 1980s before returning to 2 births in the 1990s and early 2000s.

Since the Great Recession that lasted from late 2007 until mid-2009, the U.S. total fertility rate has declined almost every year, with the exception of very small post-COVID-19 pandemic increases in 2021 and 2022. In 2024, it hit a record low, falling to 1.6. This drop is primarily driven by declines in births to people in their teens and early 20s – births that are often unintended.

But while the total fertility rate offers a snapshot of the fertility landscape, it is not a perfect indicator of how many children a woman will eventually have if fertility patterns are in flux – for example, if people are delaying having children.

Picture a 20-year-old woman today, in 2025. The total fertility rate assumes she will have the same birth rate as today’s 40-year-olds when she reaches 40. That’s not likely to be the case, because birth rates 20 years from now for 40-year-olds will almost certainly be higher than they are today, as more births occur at older ages and more people are able to overcome infertility through medically assisted reproduction.

A more nuanced picture of childbearing

These problems with the total fertility rate are why demographers also measure how many total births women have had by the end of their reproductive years. In contrast to the total fertility rate, the average number of children ever born to women ages 40 to 44 has remained fairly stable over time, hovering around two.

Americans continue to express favorable views toward childbearing. Ideal family size remains at two or more children, and 9 in 10 adults either have, or would like to have, children. However, many Americans are unable to reach their childbearing goals. This seems to be related to the high cost of raising children and growing uncertainty about the future.

In other words, it doesn’t seem to be the case that birth rates are low because people are uninterested in having children; rather, it’s because they don’t feel it’s feasible for them to become parents or to have as many children as they would like.

The challenge of predicting future population size

Standard demographic projections do not support the idea that population size is set to shrink dramatically.

One billion people lived on Earth 250 years ago. Today there are over 8 billion, and by 2100 the United Nations predicts there will be over 10 billion. That’s 2 billion more, not fewer, people in the foreseeable future. Admittedly, that projection is plus or minus 4 billion. But this range highlights another key point: Population projections get more uncertain the further into the future they extend.

Predicting the population level five years from now is far more reliable than 50 years from now – and beyond 100 years, forget about it. Most population scientists avoid making such long-term projections, for the simple reason that they are usually wrong. That’s because fertility and mortality rates change over time in unpredictable ways.

The U.S. population size is also not declining. Currently, despite fertility below the replacement level of 2.1 children per woman, there are still more births than deaths. The U.S. population is expected to grow by 22.6 million by 2050 and by 27.5 million by 2100, with immigration playing an important role.

A row of pregnant womens' torsos, no heads.
Despite a drop in fertility rates, there are still more births than deaths in the U.S.
andresr/E+ via Getty Images

Will low fertility cause an economic crisis?

A common rationale for concern about low fertility is that it leads to a host of economic and labor market problems. Specifically, pronatalists argue that there will be too few workers to sustain the economy and too many older people for those workers to support. However, that is not necessarily true – and even if it were, increasing birth rates wouldn’t fix the problem.

As fertility rates fall, the age structure of the population shifts. But a higher proportion of older adults does not necessarily mean the proportion of workers to nonworkers falls.

For one thing, the proportion of children under age 18 in the population also declines, so the number of working-age adults – usually defined as ages 18 to 64 – often changes relatively little. And as older adults stay healthier and more active, a growing number of them are contributing to the economy. Labor force participation among Americans ages 65 to 74 increased from 21.4% in 2003 to 26.9% in 2023 — and is expected to increase to 30.4% by 2033. Modest changes in the average age of retirement or in how Social Security is funded would further reduce strains on support programs for older adults.

What’s more, pronatalists’ core argument that a higher birth rate would increase the size of the labor force overlooks some short-term consequences. More babies means more dependents, at least until those children become old enough to enter the labor force. Children not only require expensive services such as education, but also reduce labor force participation, particularly for women. As fertility rates have fallen, women’s labor force participation rates have risen dramatically – from 34% in 1950 to 58% in 2024. Pronatalist policies that discourage women’s employment are at odds with concerns about a diminishing number of workers.

Research shows that economic policies and labor market conditions, not demographic age structures, play the most important role in determining economic growth in advanced economies. And with rapidly changing technologies like automation and artificial intelligence, it is unclear what demand there will be for workers in the future. Moreover, immigration is a powerful – and immediate – tool for addressing labor market needs and concerns over the proportion of workers.

Overall, there’s no evidence for Elon Musk’s assertion that “humanity is dying.” While the changes in population structure that accompany low birth rates are real, in our view the impact of these changes has been dramatically overstated. Strong investments in education and sensible economic policies can help countries successfully adapt to a new demographic reality.The Conversation

Leslie Root, Assistant Professor of Research, Institute of Behavioral Science, University of Colorado Boulder; Karen Benjamin Guzzo, Professor of Sociology and Director of the Carolina Population Center, University of North Carolina at Chapel Hill, and Shelley Clark, Professor of Sociology, McGill University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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Note: The following A.I. based commentary is not part of the original article, reproduced above, but is offered in the hopes that it will promote greater media literacy and critical thinking, by making any potential bias more visible to the reader –Staff Editor.

Political Bias Rating: Center-Left

This article presents a fact-based, analytic perspective emphasizing demographic research and economic policy over alarmist or ideological pronatalism. It critiques pronatalist views, often associated with conservative or right-leaning agendas that push for higher birth rates to support economic growth, by highlighting complexities such as women’s labor participation and immigration’s role. The language is measured, citing scholarly sources and avoiding sensationalism, reflecting a moderate but slightly progressive stance that favors evidence-based social policy and economic adaptation rather than simplistic demographic fears. The balanced tone and focus on systemic factors place it in the center-left range.

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How germy is the public pool? An infectious disease expert weighs in on poop, pee and perspiration – and the deceptive smell of chlorine

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theconversation.com – Lisa Cuchara, Professor of Biomedical Sciences, Quinnipiac University – 2025-07-23 07:39:00


On hot summer days, swimming in public pools offers great health benefits but also poses infection risks. Chlorine, commonly used in pools, kills many germs but not instantly or completely. The CDC reports over 200 pool-related illness outbreaks (2015-2019), affecting thousands with skin, respiratory, ear, and gastrointestinal infections. Hard-to-kill pathogens like Cryptosporidium can survive in chlorinated water for days, spreading through fecal contamination. Other germs include Pseudomonas aeruginosa and viruses like norovirus. Bodily fluids mix with chlorine to form potentially harmful chloramines, causing strong chlorine odors indicating contamination. To stay safe, shower before and after swimming, avoid pools when sick, take bathroom breaks, and cover wounds properly.

A 2023 CDC report tracked more than 200 pool-associated outbreaks over a four-year period. But a few basic precautions can ward off these dangers.
Maria Korneeva/Moment via Getty Images

Lisa Cuchara, Quinnipiac University

On hot summer days, few things are more refreshing than a dip in the pool. But have you ever wondered if the pool is as clean as that crystal blue water appears?

As an immunologist and infectious disease specialist, I study how germs spread in public spaces and how to prevent the spread. I even teach a course called “The Infections of Leisure” where we explore the risks tied to recreational activities and discuss precautions, while also taking care not to turn students into germophobes.

Swimming, especially in public pools and water parks, comes with its own unique set of risks — from minor skin irritations to gastrointestinal infections. But swimming also has a plethora of physical, social and mental health benefits. With some knowledge and a little vigilance, you can enjoy the water without worrying about what might be lurking beneath the surface.

The reality of pool germs

Summer news headlines and social media posts often spotlight the “ick-factor” of communal swimming spaces. These concerns do have some merit.

The good news is that chlorine, which is widely used in pools, is effective at killing many pathogens. The not-so-good news is that chlorine does not work instantly – and it doesn’t kill everything.

Every summer, the Centers for Disease Control and Prevention issues alerts about swimming-related outbreaks of illness caused by exposure to germs in public pools and water parks. A 2023 CDC report tracked over 200 pool-associated outbreaks from 2015 to 2019 across the U.S., affecting more than 3,600 people. These outbreaks included skin infections, respiratory issues, ear infections and gastrointestinal distress. Many of the outcomes from such infections are mild, but some can be serious.

Germs and disinfectants

Even in a pool that’s properly treated with chlorine, some pathogens can linger for minutes to days. One of the most common culprits is Cryptosporidium, a microscopic germ that causes watery diarrhea. This single-celled parasite has a tough outer shell that allows it to survive in chlorine-treated water for up to 10 days. It spreads when fecal matter — often from someone with diarrhea — enters the water and is swallowed by another swimmer. Even a tiny amount, invisible to the eye, can infect dozens of people.

Collection of visual symbols for pool rules
Showering before and after swimming in a public pool helps avoid both bringing in and taking out pathogens and body substances.
Hafid Firman Syarif/iStock via Getty Images Plus

Another common germ is Pseudomonas aeruginosa, a bacterium that causes hot tub rash and swimmer’s ear. Viruses like norovirus and adenovirus can also linger in pool water and cause illness.

Swimmers introduce a range of bodily residues to the water, including sweat, urine, oils and skin cells. These substances, especially sweat and urine, interact with chlorine to form chemical byproducts called chloramines that may pose health risks.

These byproducts are responsible for that strong chlorine smell. A clean pool should actually lack a strong chlorine odor, as well as any other smells, of course. It is a common myth that a strong chlorine smell is a good sign of a clean pool. In fact, it may actually be a red flag that means the opposite – that the water is contaminated and should perhaps be avoided.

How to play it safe at a public pool

Most pool-related risks can be reduced with simple precautions by both the pool staff and swimmers. And while most pool-related illnesses won’t kill you, no one wants to spend their vacation or a week of beautiful summer days in the bathroom.

These 10 tips can help you avoid germs at the pool:

  • Shower before swimming. Rinsing off for at least one minute removes most dirt and oils on the body that reduce chlorine’s effectiveness.

  • Avoid the pool if you’re sick, especially if you have diarrhea or an open wound. Germs can spread quickly in water.

  • Try to keep water out of your mouth to minimize the risk of ingesting germs.

  • Don’t swim if you have diarrhea to help prevent the spread of germs.

  • If diagnosed with cryptosporidiosis, often called “crypto,” wait two weeks after diarrhea stops before returning to the pool.

  • Take frequent bathroom breaks. For children and adults alike, regular bathroom breaks help prevent accidents in the pool.

  • Check diapers hourly and change them away from the pool to prevent fecal contamination.

  • Dry your ears thoroughly after swimming to help prevent swimmer’s ear.

  • Don’t swim with an open wound – or at least make sure it’s completely covered with a waterproof bandage to protect both you and others.

  • Shower after swimming to remove germs from your skin.The Conversation

Lisa Cuchara, Professor of Biomedical Sciences, Quinnipiac University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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The post How germy is the public pool? An infectious disease expert weighs in on poop, pee and perspiration – and the deceptive smell of chlorine appeared first on theconversation.com



Note: The following A.I. based commentary is not part of the original article, reproduced above, but is offered in the hopes that it will promote greater media literacy and critical thinking, by making any potential bias more visible to the reader –Staff Editor.

Political Bias Rating: Centrist

This content is focused on public health and safety related to swimming pools, providing factual information grounded in scientific research without promoting any political ideology. It references authoritative sources like the CDC and includes practical advice to protect public health. The language is neutral, educational, and objective, avoiding partisan framing or ideological perspectives. Overall, it reports on health risks and precautions in a balanced manner, adhering to a straightforward, informational style typical of centrist, science-based communication.

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Binary star systems are complex astronomical objects − a new AI approach could pin down their properties quickly

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theconversation.com – Andrej Prša, Professor of Astrophysics and Planetary Science, Villanova University – 2025-07-23 07:38:00


Stars are cosmic building blocks, often found in binary systems where two stars orbit a common center, governed by Kepler’s law, allowing astronomers to measure their combined mass. Radii are determined using eclipsing binaries—systems aligned so one star passes in front of another, enabling radius calculation through eclipse shapes. Modeling these systems is complex and extremely time-consuming due to star interactions and shapes, requiring extensive computations. To address this, researchers developed deep-learning neural networks trained on vast simulated data to rapidly predict binary star observations, achieving a millionfold speed increase. This AI-driven method matches physical models with over 99% accuracy, enabling quicker analysis of stellar properties across millions of binaries and promising broader applications.

In a binary star system, two stars orbit around each other.
ESO/L. Calçada, CC BY

Andrej Prša, Villanova University

Stars are the fundamental building blocks of our universe. Most stars host planets, like our Sun hosts our solar system, and if you look more broadly, groups of stars make up huge structures such as clusters and galaxies. So before astrophysicists can attempt to understand these large-scale structures, we first need to understand basic properties of stars, such as their mass, radius and temperature.

But measuring these basic properties has proved exceedingly difficult. This is because stars are quite literally at astronomical distances. If our Sun were a basketball on the East Coast of the U.S., then the closest star, Proxima, would be an orange in Hawaii. Even the world’s largest telescopes cannot resolve an orange in Hawaii. Measuring radii and masses of stars appears to be out of scientists’ reach.

Enter binary stars. Binaries are systems of two stars revolving around a mutual center of mass. Their motion is governed by Kepler’s harmonic law, which connects three important quantities: the sizes of each orbit, the time it takes for them to orbit, called the orbital period, and the total mass of the system.

I’m an astronomer, and my research team has been working on advancing our theoretical understanding and modeling approaches to binary stars and multiple stellar systems. For the past two decades we’ve also been pioneering the use of artificial intelligence in interpreting observations of these cornerstone celestial objects.

Measuring stellar masses

Astronomers can measure orbital size and period of a binary system easily enough from observations, so with those two pieces they can calculate the total mass of the system. Kepler’s harmonic law acts as a scale to weigh celestial bodies.

An animation of a large star, which appears stationary, with a smaller, brighter star orbiting around it and eclipsing it when it passes in front.
Binary stars orbit around each other, and in eclipsing binary stars, one passes in front of the other, relative to the telescope lens.
Merikanto/Wikimedia Commons, CC BY-SA

Think of a playground seesaw. If the two kids weigh about the same, they’ll have to sit at about the same distance from the midpoint. If, however, one child is bigger, he or she will have to sit closer, and the smaller kid farther from the midpoint.

It’s the same with stars: The more massive the star in a binary pair, the closer to the center it is and the slower it revolves about the center. When astronomers measure the speeds at which the stars move, they can also tell how large the stars’ orbits are, and as a result, what they must weigh.

Measuring stellar radii

Kepler’s harmonic law, unfortunately, tells astronomers nothing about the radii of stars. For those, astronomers rely on another serendipitous feature of Mother Nature.

Binary star orbits are oriented randomly. Sometimes, it happens that a telescope’s line of sight aligns with the plane a binary star system orbits on. This fortuitous alignment means the stars eclipse one another as they revolve about the center. The shapes of these eclipses allow astronomers to find out the stars’ radii using straightforward geometry. These systems are called eclipsing binary stars.

By taking measurements from an eclipsing binary star system, astronomers can measure the radii of the stars.

More than half of all Sun-like stars are found in binaries, and eclipsing binaries account for about 1% to 2% of all stars. That may sound low, but the universe is vast, so there are lots and lots of eclipsing systems out there – hundreds of millions in our galaxy alone.

By observing eclipsing binaries, astronomers can measure not only the masses and radii of stars but also how hot and how bright they are.

Complex problems require complex computing

Even with eclipsing binaries, measuring the properties of stars is no easy task. Stars are deformed as they rotate and pull on each other in a binary system. They interact, they irradiate one another, they can have spots and magnetic fields, and they can be tilted this way or that.

To study them, astronomers use complex models that have many knobs and switches. As an input, the models take parameters – for example, a star’s shape and size, its orbital properties, or how much light it emits – to predict how an observer would see such an eclipsing binary system.

Computer models take time. Computing model predictions typically takes a few minutes. To be sure that we can trust them, we need to try lots of parameter combinations – typically tens of millions.

This many combinations requires hundreds of millions of minutes of compute time, just to determine basic properties of stars. That amounts to over 200 years of computer time.

Computers linked in a cluster can compute faster, but even using a computer cluster, it takes three or more weeks to “solve,” or determine all the parameters for, a single binary. This challenge explains why there are only about 300 stars for which astronomers have accurate measurements of their fundamental parameters.

The models used to solve these systems have already been heavily optimized and can’t go much faster than they already do. So, researchers need an entirely new approach to reducing computing time.

Using deep learning

One solution my research team has explored involves deep-learning neural networks. The basic idea is simple: We wanted to replace a computationally expensive physical model with a much faster AI-based model.

First, we computed a huge database of predictions about a hypothetical binary star – using the features that astronomers can readily observe – where we varied the hypothetical binary star’s properties. We are talking hundreds of millions of parameter combinations. Then, we compared these results to the actual observations to see which ones best match up. AI and neural networks are ideally suited for this task.

In a nutshell, neural networks are mappings. They map a certain known input to a given output. In our case, they map the properties of eclipsing binaries to the expected predictions. Neural networks emulate the model of a binary but without having to account for all the complexity of the physical model.

Neural networks detect patterns and use their training to predict an output, based on an input.

We train the neural network by showing it each prediction from our database, along with the set of properties used to generate it. Once fully trained, the neural network will be able to accurately predict what astronomers should observe from the given properties of a binary system.

Compared to a few minutes of runtime for the physical model, a neural network uses artificial intelligence to get the same result within a tiny fraction of a second.

Reaping the benefits

A tiny fraction of a second works out to about a millionfold runtime reduction. This brings the time down from weeks on a supercomputer to mere minutes on a single laptop. It also means that we can analyze hundreds of thousands of binary systems in a couple of weeks on a computer cluster.

This reduction means we can obtain fundamental properties – stellar masses, radii, temperatures and luminosities – for every eclipsing binary star ever observed within a month or two. The big challenge remaining is to show that AI results really give the same results as the physical model.

This task is the crux of my team’s new paper. In it we’ve shown that, indeed, the AI-driven model yields the same results as the physical model across over 99% of parameter combinations. This result means the AI’s performance is robust. Our next step? Deploy the AI on all observed eclipsing binaries.

Best of all? While we applied this methodology to binaries, the basic principle applies to any complex physical model out there. Similar AI models are already speeding up many real-world applications, from weather forecasting to stock market analysis.The Conversation

Andrej Prša, Professor of Astrophysics and Planetary Science, Villanova University

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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Note: The following A.I. based commentary is not part of the original article, reproduced above, but is offered in the hopes that it will promote greater media literacy and critical thinking, by making any potential bias more visible to the reader –Staff Editor.

Political Bias Rating: Centrist

This article presents a factual and technical explanation of astrophysics research without any ideological framing or political language. It focuses on scientific concepts, methods, and advancements in measuring star properties using AI, maintaining a neutral, educational tone throughout. The content neither promotes nor criticizes political viewpoints, policies, or social issues, and therefore adheres to objective reporting on a scientific topic. The emphasis is on innovation and research progress, free of partisan influence or bias.

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