Prime numbers, the building blocks of mathematics, have fascinated for centuries − now technology is revolutionizing the search for them
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.
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 Euler proved that there are infinitely many prime numbers. Euler 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, Euler 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. Euler 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.
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 11 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 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.
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.
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 article is republished from The Conversation, a nonprofit, independent news organization bringing you facts and trustworthy analysis to help you make sense of our complex world. It was written by: Jeremiah Bartz, University of North Dakota
Read more:
Planning the best route with multiple destinations is hard even for supercomputers – a new approach breaks a barrier that's stood for nearly half a century
Why does nature create patterns? A physicist explains the molecular-level processes behind crystals, stripes and basalt columns
Art and science illuminate the same subtle proportions in tree branches
Jeremiah Bartz owns shares in Nvidia.
Try Our AI Features
Explore what Daily8 AI can do for you:
Comments
No comments yet...
Related Articles
Business Insider
11 hours ago
- Business Insider
How Allegro MicroSystems (ALGM) Can Outpace AMD in Boosting Returns
With the advent of artificial intelligence, it's no surprise that semiconductor giants like Advanced Micro Devices (AMD) have surged in value in recent years. However, the company isn't immune to pressure, as its latest financial disclosure demonstrated, with statistic hot-shots like Allegro MicroSystems (ALGM) waiting in the wings. Elevate Your Investing Strategy: Take advantage of TipRanks Premium at 50% off! Unlock powerful investing tools, advanced data, and expert analyst insights to help you invest with confidence. Thanks to investors increasingly ramping up their expectations, hiccups against key benchmark figures can result in tremendous volatility, as we've just witnessed in AMD stock. For the second quarter, Advanced Micro reported record revenue of $7.7 billion, representing a 32% jump against the year-ago period. Fundamentally, much of the growth could be attributed to strong demand for server and PC processors. Not surprisingly, the latest sales tally beat Wall Street analysts' consensus view of $7.41 billion. Unfortunately, the company couldn't sustain the same magnitude of momentum on the bottom line. Adjusted earnings per share landed in line with the consensus estimate. However, this figure represented a huge decline of 30.4% on a year-over-year basis. In terms of outlook, management guided Q3 revenue to reach approximately $8.7 billion, reflecting 28% year-over-year growth. Notably, this forecast swung past the Street's estimate of $8.3 billion. Still, it wasn't enough to prevent an initial panic in AMD stock following the disclosure. That said, the equity entered this past weekend on a high note. With the bulls once again demonstrating their resilience, AMD stock may look interesting to options traders. However, there's another overlooked chipmaker that astute speculators may want to consider. AMD's Shining Moment Has Passed To be 100% clear, I'm not trashing AMD stock. If you genuinely believe that the security offers viable upside, I won't protest. However, the point I'd like to make is that based on a statistical framework, AMD lacks a confidence-inspiring signal. In the options arena, the financial publication industry focuses heavily on convoluted concepts such as implied volatility, unusual activity, and the Greeks. While these are certainly intriguing datapoints, that's all they are. It's like looking at the speedometer — useful, perhaps, to avoid getting a traffic ticket, but having this knowledge alone won't make you drive like Max Verstappen. Here's the ultimate truth when it comes to options: it doesn't matter how cheap your option is if the probabilistic risk of failure is sky-high. No amount of Greek fluency can save a trade destined for the dumps. As such, I'm a big believer in discrete-event analysis. Rather than view share prices in their native form as continuous scalar signals, I prefer to convert and categorize price action into discrete, defined states. For example, in the past 10 weeks, I'm no longer interested in how AMD stock transitioned from one price point to another. Instead, I look at how the market voted to buy AMD seven times and sell three times. During this period, the security enjoyed an upward trajectory. For brevity, we can label this sequence as 7-3-U. For a deeper delve into my proprietary quantitative analysis and how 'sequences' are derived, readers can refer to previous articles. The goal of categorizing these 10-week sequences is to use past analogs to determine how the market responds to them. As a baseline, the chance that a long position in AMD stock will rise on any given week is 55.07%. This is effectively our null hypothesis, the probabilistic expectation assuming no mispricing. However, our alternative hypothesis is that, because of the 7-3-U sequence, the odds have changed. Sadly, the odds change to the projected detriment of bullish traders, with the probability of upside shrinking to 40.43%. This translates to a downside probability of 59.57%, which, in my opinion, isn't enough to justify an outright bearish position. From a trader's perspective, AMD stock is stuck in no-man's land. Allegro MicroSystems: The Arguably Better Alternative For those seeking speculative growth, I would much rather consider Allegro MicroSystems. A semiconductor specialist, Allegro designs, develops, manufactures, and markets sensor and power integrated circuits (ICs) that are used in a variety of applications, especially in the automotive and industrial sectors. While ALGM stock doesn't command the footprint of an AMD, it's a name respected by several market experts. Since the start of the year, ALGM has gained nearly 46%, which is an impressive performance. However, in the past month, the security has been down roughly 17%, thereby presenting a discounted opportunity for aggressive traders. It's not just about the red ink; rather, the statistical profile is very compelling. In the last 10 weeks, the market voted to buy ALGM stock six times and sell four times. During this period, the security swung upward. Again, for brevity, we can label this sequence as 6-4-U. Here, the baseline probability — or the null hypothesis — is 50.8%, barely a positive bias. However, with the 6-4-U sequence flashing, our alternative hypothesis stands at 63.89%. Therefore, an incentive exists to consider a debit-based options strategy. From past analogs, the median expected performance assuming the positive pathway is 4.88% in the following week when the 6-4-U sequence flashes. If the bulls manage to maintain control of the market for the next three weeks, traders may anticipate an additional performance boost of 1.41%. With ALGM stock closing at $30.77 on Friday, the upside target could be $32.73, possibly $33 if the winds move favorably. Thanks to the structure of the options market right now, a compelling idea may be buying the $30 call expiring September 19 outright. Presently, the ask on this contract is $2.50 or $250 when applying the options multiplier. Based on intrinsic value at expiration, the breakeven price for this trade is $32.50. Another trade to consider for extreme speculators is the 35/40 bull call spread expiring on November 21. This transaction involves buying the $35 call and simultaneously selling the $40 call, for a net debit paid of $120 (the maximum possible loss). Should ALGM stock rise through the short strike price of $40 at expiration, the maximum profit is $380, a payout of about 317%. Keep in mind that looking that far ahead is difficult for any forecasting model to get right. However, I will say that the p-value (from a one-tailed binomial test) of the 6-4-U sequence is 0.1353. This means that there's a 13.53% chance that the implications of the signal could materialize randomly as opposed to intentionally. I can't call this a statistically significant signal, but it remains an empirically intriguing sequence within the stock market's open and entropic system. Is ALGM a Good Stock to Buy? Turning to Wall Street, ALGM stock carries a Strong Buy consensus rating based on nine Buys, two Holds, and zero Sell ratings over the past three months. ALGM's average stock price target is $38.09, implying almost 20% upside potential. Allegro Offers a Superior Options Trading Opportunity Than AMD While Advanced Micro Devices has been a fan favorite, there's growing evidence that traders are getting stretched with AMD stock, putting the security in ambiguous air. In contrast, ALGM stock offers a much clearer probabilistic signal, making it an intriguing alternative.
Fox News
2 days ago
- Fox News
Researchers discover 2,500-year-old honey residue in ancient bronze jars
Researchers recently made a sticky discovery. They found 2,500-year-old honey, sealed in a vessel and buried underground for nearly three millennia. The residue was found in bronze jars at an underground shrine in Paestum, an ancient Greek settlement in modern-day Campania, Italy. The artifacts date back to the 6th century B.C. The jars were first found by archaeologists in 1954, but the residue in them has eluded experts – until now. In a study published by the American Chemical Society on July 30, experts concluded that the waxy residue was once honey. Luciana Carvalho, a research associate at the University of Oxford, told Fox News Digital the substance bears little resemblance to the golden honey in most modern-day cupboards. "Ancient honey was quite different from the clear, smooth honey we buy today," she noted. "There was no ultra‑filtration, no pasteurization and no synthetic pesticides in the landscape." The chemical results "strongly" suggested that the jars held raw honeycomb, rather than a blended product like most modern honey. "If honey had been mixed with milk, for example, we would expect to see extra fatty acids in the residue – and we don't," she said. But the raw honeycomb has dramatically changed over the millennia. Along with fellow researchers Elisabete Pires and James McCullagh, Carvalho found that the sugar gradually became dark and acidic. "After 2,500 years, almost all the original sugars have broken down [and been] eaten by microbes, so the residue isn't sweet anymore." "[The] sugars slowly reacted with proteins in a kind of slow-motion browning process, similar to what happens when bread bakes, turning it darker and more acidic," she said. Because of that, the remaining residue isn't exactly pleasant to eat, according to the expert. "What survives now is a waxy residue with a slight tang and virtually no sweetness," Carvalho noted. "After 2,500 years, almost all the original sugars have broken down [and been] eaten by microbes, so the residue isn't sweet anymore." As for the vessels, Carvalho said the bronze jars have cork discs that seal their necks, which points to "careful storage of something valuable." The copper-alloy jugs were found in a sealed, underground shrine, suggesting that they were left there as part of a ritual. "Inside, the residue clung to the bottoms and sides, exactly what you'd expect if raw honeycomb had been placed inside and slowly dried out over centuries," she said. "We hope our approach will be used to identify honey residues in other museum collections so we can learn more about ancient beekeeping and the role of honey in diet, medicine and ritual life." Researchers in the 1980s previously believed that the honey was a mixture of wax, fat and resin, with Carvalho noting that past research was limited by less precise tools. "Those methods were great for detecting fats and waxes but couldn't easily pick up sugars or proteins without extra chemical steps," she said. "In our study we used multiple modern techniques designed to detect different types of molecules, including sugars and proteins, even if these are present at trace levels, with instruments far more sensitive than anything available in the 1980s." She also noted that the discovery had strong collaboration from multiple groups, including museum curators, conservators and specialized scientists. "We hope our approach will now be used to identify honey residues in other museum collections so we can learn more about ancient beekeeping and the role of honey in diet, medicine and ritual life," Carvalho said. The latest research adds to a number of ancient food-related discoveries this year, which are extremely rare occurrences. In Guam, 3,500-year-old rice was recently found, making it the earliest known evidence of rice in Remote Oceania. Earlier in 2025, archaeologists uncovered a well-preserved loaf of ancient bread in Turkey, dating back to the Bronze Age.
Yahoo
3 days ago
- Yahoo
EU clears €4.1bn Just Eat takeover but prevents food delivery mega-merger
The European Commission has approved the €4.1 billion acquisition of Just Eat (JET) by Naspers, through its investment arm Prosus, after the technology group agreed to take concrete steps to reduce its influence in Europe's food delivery market. Prosus already holds a significant minority stake in Delivery Hero—one of the largest food delivery companies in Austria, Bulgaria, Italy, Poland and Spain—which operates popular brands such as Glovo, Foodora, and efood. The Commission said the measures were necessary to prevent Naspers from having significant sway over two of the sector's largest competitors, which could weaken competition and harm consumers. "To address the Commission's concerns, Naspers offered to significantly reduce its shareholding in Delivery Hero, below a specified very low percentage, within 12 months and to implement a set of additional commitments," the EU body said in a statement. "The Commission has the duty to assess mergers and acquisitions involving companies with a turnover above certain thresholds," the statement continued. Why is this important? In June 2025, the European Commission fined Delivery Hero and Glovo €329 million for operating a cartel—a high market concentration that lets them agree to drive up prices while not improving services—in the food delivery sector. That case reinforced regulators' concerns that without strict safeguards, large players may coordinate markets for profit instead of competing with both prices and quality. Related Chipmakers Nvidia and AMD to pay 15% of China revenue to US government Greek ban on ATM charges and limits on other banking fees comes into force today Just Eat runs familiar delivery platforms regularly used in Europe, such as Just Eat, Lieferando and others. Between them, these companies would control a significant slice of Europe's meal delivery market, which means most restaurants and customers would have limited alternatives. The European online food delivery market is worth tens of billions annually and has grown rapidly since the pandemic, making it a critical part of urban economies. This market has few big players, so any merger or ownership overlap can have an outsized impact on prices, restaurant commission rates and courier pay. The €329 million fine against Delivery Hero and Glovo in June 2025 proves that colluding within the market is not a theoretical risk and that European regulators have already caught major players gaming the market to limit competition. The deal is particularly notable as it is unusual for the Commission to demand a major sell-off of shares in another company or existing asset during a Phase I review, as it did on Monday. This signals that Brussels is taking a harder line on digital platform consolidation, especially in markets where just two or three players dominate across the bloc.
