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

Prime numbers, essential in mathematics and history, have been studied since ancient times. They are defined as whole numbers greater than one with only two divisors: one and themselves.

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Last Updated on August 31, 2025 by Daily News Staff

file 20250520 56 lrfedu.png?ixlib=rb 4.1

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.

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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.

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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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Urbanism

The Building That Proved Los Angeles Could Go Vertical

Los Angeles once banned skyscrapers, yet City Hall broke the height limit and proved high-rise buildings could be engineered safely in an earthquake zone.

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Last Updated on February 19, 2026 by Daily News Staff

Los Angeles once banned skyscrapers, yet City Hall broke the height limit and proved high-rise buildings could be engineered safely in an earthquake zone.
LA City Hall. Image Credit: TNC Network & Envato

How City Hall Quietly Undermined LA’s Own Height Limits

The Knowledge Series | STM Daily News

For more than half a century, Los Angeles enforced one of the strictest building height limits in the United States. Beginning in 1905, most buildings were capped at 150 feet, shaping a city that grew outward rather than upward.

The goal was clear: avoid the congestion, shadows, and fire dangers associated with dense Eastern cities. Los Angeles sold itself as open, sunlit, and horizontal — a place where growth spread across land, not into the sky.

And yet, in 1928, Los Angeles City Hall rose to 454 feet, towering over the city like a contradiction in concrete.

It wasn’t built to spark a commercial skyscraper boom.
But it ended up proving that Los Angeles could safely build one.


A Rule Designed to Prevent a Manhattan-Style City

The original height restriction was rooted in early 20th-century fears:

  • Limited firefighting capabilities
  • Concerns over blocked sunlight and airflow
  • Anxiety about congestion and overcrowding
  • A strong desire not to resemble New York or Chicago

Los Angeles wanted prosperity — just not vertical density.

The height cap reinforced a development model where:

  • Office districts stayed low-rise
  • Growth moved outward
  • Automobiles became essential
  • Downtown never consolidated into a dense core

This philosophy held firm even as other American cities raced upward.


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Los Angeles banned skyscrapers for decades — except one. 🏛️ While most buildings were capped at 150 feet, LA City Hall rose three times higher. This wasn’t a loophole — it was power, symbolism, and city planning shaping the skyline we know today. Why was City Hall the exception? And how did this one decision change Los Angeles forever? 📍 Forgotten LA 🧠 The Knowledge Series 📰 STM Daily News LosAngelesHistory LACityHall ForgottenLA UrbanPlanning CityPlanning LASkyline DidYouKnow HistoryTok TheKnowledge STMDailyNews ♬ original sound – STMDailyNews – STMDailyNews


Why City Hall Was Never Meant to Change the Rules

City Hall was intentionally exempt from the height limit because the law applied primarily to private commercial buildings, not civic monuments.

But city leaders were explicit about one thing:
City Hall was not a precedent.

It was designed to:

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  • Serve as a symbolic seat of government
  • Stand alone as a civic landmark
  • Represent stability, authority, and modern governance
  • Avoid competing with private office buildings

In effect, Los Angeles wanted a skyline icon — without a skyline.


Innovation Hidden in Plain Sight

What made City Hall truly significant wasn’t just its height — it was how it was built.

At a time when seismic science was still developing, City Hall incorporated advanced structural ideas for its era:

  • A steel-frame skeleton designed for flexibility
  • Reinforced concrete shear walls for lateral strength
  • A tapered tower to reduce wind and seismic stress
  • Thick structural cores that distributed force instead of resisting it rigidly

These choices weren’t about aesthetics — they were about survival.


The Earthquake That Changed the Conversation

In 1933, the Long Beach earthquake struck Southern California, causing widespread damage and reshaping building codes statewide.

Los Angeles City Hall survived with minimal structural damage.

This moment quietly reshaped the debate:

  • A tall building had endured a major earthquake
  • Structural engineering had proven effective
  • Height alone was no longer the enemy — poor design was

City Hall didn’t just survive — it validated a new approach to vertical construction in seismic regions.


Proof Without Permission

Despite this success, Los Angeles did not rush to repeal its height limits.

Cultural resistance to density remained strong, and developers continued to build outward rather than upward. But the technical argument had already been settled.

City Hall stood as living proof that:

  • High-rise buildings could be engineered safely in Los Angeles
  • Earthquakes were a challenge, not a barrier
  • Fire, structural, and seismic risks could be managed

The height restriction was no longer about safety — it was about philosophy.


The Ironic Legacy

When Los Angeles finally lifted its height limit in 1957, the city did not suddenly erupt into skyscrapers. The habit of building outward was already deeply entrenched.

The result:

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  • A skyline that arrived decades late
  • Uneven density across the region
  • Multiple business centers instead of one core
  • Housing and transit challenges baked into the city’s growth pattern

City Hall never triggered a skyscraper boom — but it quietly made one possible.


Why This Still Matters

Today, Los Angeles continues to wrestle with:

  • Housing shortages
  • Transit-oriented development debates
  • Height and zoning battles near rail corridors
  • Resistance to density in a growing city

These debates didn’t begin recently.

They trace back to a single contradiction: a city that banned tall buildings — while proving they could be built safely all along.

Los Angeles City Hall wasn’t just a monument.
It was a test case — and it passed.

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The Long Track Back

Why Downtown Los Angeles Feels Small Compared to Other Cities

Downtown Los Angeles often feels “small” compared to other U.S. cities, but that’s only part of the story. With some of the tallest buildings west of the Mississippi and skyline clusters spread across the region, LA’s downtown reflects the city’s unique polycentric identity—one that, if combined, could form a true mega downtown.

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Last Updated on February 18, 2026 by Daily News Staff

Downtown Los Angeles

Panorama of Los Angeles from Mount Hollywood – California, United States

When people think of major American cities, they often imagine a bustling, concentrated downtown core filled with skyscrapers. New York has Manhattan, Chicago has the Loop, San Francisco has its Financial District. Los Angeles, by contrast, often leaves visitors surprised: “Is this really downtown?”

The answer is yes—and no.

Downtown LA in Context

Compared to other major cities, Downtown Los Angeles (DTLA) is relatively small as a central business district. For much of the 20th century, strict height restrictions capped most buildings under 150 feet, while cities like Chicago and New York were erecting early skyscrapers. LA’s skyline didn’t really begin to climb until the late 1960s.

But history alone doesn’t explain why DTLA feels different. The real story lies in how Los Angeles grew: not as one unified city center, but as a collection of many hubs.

Downtown Los Angeles

Downtown Los Angeles

A Polycentric City

Los Angeles is famously decentralized. Hollywood developed around the film industry. Century City rose on former studio land as a business hub. Burbank became a studio and aerospace center. Long Beach grew around the port. The Wilshire Corridor filled with office towers and condos.

Unlike other cities where downtown is the place for work, culture, and finance, Los Angeles spread its energy outward. Freeways and car culture made it easy for businesses and residents to operate outside of downtown. The result is a polycentric metropolis, with multiple “downtowns” rather than one dominant core.

A Resident’s Perspective

As someone who lived in Los Angeles for 28 years, I see DTLA differently. While some outsiders describe it as “small,” the reality is that Downtown Los Angeles is still significant. It has some of the tallest buildings west of the Mississippi River, including the Wilshire Grand Center and the U.S. Bank Tower. Over the last two decades, adaptive reuse projects have transformed old office buildings into lofts, while developments like LA Live, Crypto.com Arena, and the Broad Museum have revitalized the area.

In other words, DTLA is large enough—it just plays a different role than downtowns in other American cities.

Downtown Los Angeles

View of Westwood, Century City, Beverly Hills, and the Wilshire Corridor.

The “Mega Downtown” That Isn’t

A friend once put it to me with a bit of imagination: “If you could magically pick up all of LA’s skyline clusters—Downtown, Century City, Hollywood, the Wilshire Corridor—and drop them together in one spot, you’d have a mega downtown.”

He’s right. Los Angeles doesn’t lack tall buildings or urban energy—it just spreads them out over a vast area, reflecting the city’s unique history, geography, and culture.

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A Downtown That Fits Its City

So, is Downtown LA “small”? Compared to Manhattan or Chicago’s Loop, yes. But judged on its own terms, DTLA is a vibrant hub within a much larger, decentralized metropolis. It’s a downtown that reflects Los Angeles itself: sprawling, diverse, and impossible to fit neatly into the mold of other American cities.

🔗 Related Links

Dive into “The Knowledge,” where curiosity meets clarity. This playlist, in collaboration with STMDailyNews.com, is designed for viewers who value historical accuracy and insightful learning. Our short videos, ranging from 30 seconds to a minute and a half, make complex subjects easy to grasp in no time. Covering everything from historical events to contemporary processes and entertainment, “The Knowledge” bridges the past with the present. In a world where information is abundant yet often misused, our series aims to guide you through the noise, preserving vital knowledge and truths that shape our lives today. Perfect for curious minds eager to discover the ‘why’ and ‘how’ of everything around us. Subscribe and join in as we explore the facts that matter.  https://stmdailynews.com/the-knowledge/

 

 

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The Knowledge

How a 22-year-old George Washington learned how to lead, from a series of mistakes in the Pennsylvania wilderness

This Presidents Day, I’ve been thinking about George Washington − not at his finest hour, but possibly at his worst.

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How a 22-year-old George Washington learned how to lead, from a series of mistakes in the Pennsylvania wilderness
A young George Washington was thrust into the dense, contested wilderness of the Ohio River Valley as a land surveyor for real estate development companies in Virginia. Henry Hintermeister/Public domain via Wikimedia Commons

Christopher Magra, University of Tennessee

This Presidents Day, I’ve been thinking about George Washington − not at his finest hour, but possibly at his worst.

In 1754, a 22-year-old Washington marched into the wilderness surrounding Pittsburgh with more ambition than sense. He volunteered to travel to the Ohio Valley on a mission to deliver a letter from Robert Dinwiddie, governor of Virginia, to the commander of French troops in the Ohio territory. This military mission sparked an international war, cost him his first command and taught him lessons that would shape the American Revolution.

As a professor of early American history who has written two books on the American Revolution, I’ve learned that Washington’s time spent in the Fort Duquesne area taught him valuable lessons about frontier warfare, international diplomacy and personal resilience.

The mission to expel the French

In 1753, Dinwiddie decided to expel French fur trappers and military forces from the strategic confluence of three mighty waterways that crisscrossed the interior of the continent: the Allegheny, Monongahela and Ohio rivers. This confluence is where downtown Pittsburgh now stands, but at the time it was wilderness.

King George II authorized Dinwiddie to use force, if necessary, to secure lands that Virginia was claiming as its own.

As a major in the Virginia provincial militia, Washington wanted the assignment to deliver Dinwiddie’s demand that the French retreat. He believe the assignment would secure him a British army commission.

Washington received his marching orders on Oct. 31, 1753. He traveled to Fort Le Boeuf in northwestern Pennsylvania and returned a month later with a polite but firm “no” from the French.

A close-up portrait of a young, brunette George Washington.
George Washington held an honorary commission as a major in the British army prior to the French and Indian War. Dea/M. Seemuller/De Agostini collection/Getty Images

Dinwiddie promoted Washington from major to lieutenant colonel and ordered him to return to the Ohio River Valley in April 1754 with 160 men. Washington quickly learned that French forces of about 500 men had already constructed the formidable Fort Duquesne at the forks of the Ohio. It was at this point that he faced his first major test as a military leader. Instead of falling back to gather more substantial reinforcements, he pushed forward. This decision reflected an aggressive, perhaps naive, brand of leadership characterized by a desire for action over caution.

Washington’s initial confidence was high. He famously wrote to his brother that there was “something charming” in the sound of whistling bullets.

The Jumonville affair and an international crisis

Perhaps the most controversial moment of Washington’s early leadership occurred on May 28, 1754, about 40 miles south of Fort Duquesne. Guided by the Seneca leader Tanacharison – known as the “Half King” – and 12 Seneca warriors, Washington and his detachment of 40 militiamen ambushed a party of 35 French Canadian militiamen led by Ensign Joseph Coulon de Jumonville. The Jumonville affair lasted only 15 minutes, but its repercussions were global.

A color illustration showing battle between soldiers in red and blue coats.
The Jumonville affair became the opening battle of the French and Indian War. Interim Archives/Archive Collection/Getty Images

Ten of the French, including Jumonville, were killed. Washington’s inability to control his Native American allies – the Seneca warriors executed Jumonville – exposed a critical gap in his early leadership. He lacked the ability to manage the volatile intercultural alliances necessary for frontier warfare.

Washington also allowed one enemy soldier to escape to warn Fort Duquesne. This skirmish effectively ignited the French and Indian War, and Washington found himself at the center of a burgeoning international crisis.

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Defeat at Fort Necessity

Washington then made the fateful decision to dig in and call for reinforcements instead of retreating in the face of inevitable French retaliation. Reinforcements arrived: 200 Virginia militiamen and 100 British regulars. They brought news from Dinwiddie: congratulations on Washington’s victory and his promotion to colonel.

His inexperience showed in his design of Fort Necessity. He positioned the small, circular palisade in a meadow depression, where surrounding wooded high ground allowed enemy marksmen to fire down with impunity. Worse still, Tanacharison, disillusioned with Washington’s leadership and the British failure to follow through with promised support, had already departed with his warriors weeks earlier. When the French and their Native American allies finally attacked on July 3, heavy rains flooded the shallow trenches, soaking gunpowder and leaving Washington’s men vulnerable inside their poorly designed fortification.

A black and white illustration showing George Washington signing a document.
Washington was outnumbered and outmaneuvered at Fort Necessity. Interim Archives/Archive Collection/Getty Images

The battle of Fort Necessity was a grueling, daylong engagement in the mud and rain. Approximately 700 French and Native American allies surrounded the combined force of 460 Virginian militiamen and British regulars. Despite being outnumbered and outmaneuvered, Washington maintained order among his demoralized troops. When French commander Louis Coulon de Villiers – Jumonville’s brother – offered a truce, Washington faced the most humbling moment of his young life: the necessity of surrender. His decision to capitulate was a pragmatic act of leadership that prioritized the survival of his men over personal honor.

The surrender also included a stinging lesson in the nuances of diplomacy. Because Washington could not read French, he signed a document that used the word “l’assassinat,” which translates to “assassination,” to describe Jumonville’s death. This inadvertent admission that he had ordered the assassination of a French diplomat became propaganda for the French, teaching Washington the vital importance of optics in international relations.

A current photograph of the logs used to construct Fort Necessity as it stands today along the battlefield in Pennsylvania.
A log cabin used to protect the perishable supplies still stands at Fort Necessity today. MyLoupe/Universal Images Group/Getty Images

Lessons that forged a leader

The 1754 campaign ended in a full retreat to Virginia, and Washington resigned his commission shortly thereafter. Yet, this period was essential in transforming Washington from a man seeking personal glory into one who understood the weight of responsibility.

He learned that leadership required more than courage – it demanded understanding of terrain, cultural awareness of allies and enemies, and political acumen. The strategic importance of the Ohio River Valley, a gateway to the continental interior and vast fur-trading networks, made these lessons all the more significant.

Ultimately, the hard lessons Washington learned at the threshold of Fort Duquesne in 1754 provided the foundational experience for his later role as commander in chief of the Continental Army. The decisions he made in Pennsylvania and the Ohio wilderness, including the impulsive attack, the poor choice of defensive ground and the diplomatic oversight, were the very errors he would spend the rest of his military career correcting.

Though he did not capture Fort Duquesne in 1754, the young George Washington left the woods of Pennsylvania with a far more valuable prize: the tempered, resilient spirit of a leader who had learned from his mistakes.

Christopher Magra, Professor of American History, University of Tennessee

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

 
Dive into “The Knowledge,” where curiosity meets clarity. This playlist, in collaboration with STMDailyNews.com, is designed for viewers who value historical accuracy and insightful learning. Our short videos, ranging from 30 seconds to a minute and a half, make complex subjects easy to grasp in no time. Covering everything from historical events to contemporary processes and entertainment, “The Knowledge” bridges the past with the present. In a world where information is abundant yet often misused, our series aims to guide you through the noise, preserving vital knowledge and truths that shape our lives today. Perfect for curious minds eager to discover the ‘why’ and ‘how’ of everything around us. Subscribe and join in as we explore the facts that matter.  https://stmdailynews.com/the-knowledge/
 

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