Latest news with #Babylonian
Yahoo
3 days ago
- Science
- Yahoo
Prime numbers, the building blocks of mathematics, have fascinated for centuries − now technology is revolutionizing the search for them
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. 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.
UPI
3 days ago
- Science
- UPI
Tech revolutionizes centuries-old search for prime numbers
A laptop represents today's tools fueling the global search for massive prime numbers, essential for both mathematics and cybersecurity. Photo by Tony Avelar/EPA May 30 (UPI) -- 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 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 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. 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. Jeremiah Bartz is an associate professor of mathematics at University of North Dakota. This article is republished from The Conversation under a Creative Commons license. Read the original article. The views and opinions in this commentary are solely those of the author.
Shafaq News
3 days ago
- General
- Shafaq News
Erbil Citadel shines again with weekly public access
Shafaq News/ Erbil's historic citadel reopened to the public for one day a week as its large-scale restoration project approaches completion. The Head of the Citadel Rehabilitation Commission, Falah Hassan told Shafaq News that 1,200 cubic meters of the citadel's flooring have been paved over the past six months, along with the rehabilitation of the main street connecting the Ahmadia Gate to the Grand Bazaar Gate. The restoration of Erbil Citadel is entering its final phase, with technical issues set to be resolved within days, and work on the electrical, sewage, bathhouse, and mosque nearing completion, Hassan said, confirming that the site is now illuminated and open weekly for public visits while restoration continues in other areas. He further pointed out that the management of the UNESCO World Heritage landmarks was officially transferred from Erbil's provincial authority to the Kurdistan Regional Government's Cabinet in recent months. The move streamlines decision-making and allows requests to be submitted directly to the Council of Ministers. Erbil Citadel was added to UNESCO's World Heritage List in 2014 for its unique architectural significance. Historical records trace its origins through multiple civilizations, including the Sumerian, Akkadian, Babylonian, Assyrian, Persian, Greek, Islamic, and Ottoman eras. The site includes three main neighborhoods, traditional homes, mosques, Sufi lodges, and public baths. Its distinctive circular wall is among its most prominent features, known for its durability and design.
Iraqi News
5 days ago
- General
- Iraqi News
Iraqis restore ancient tomb of Rabbi Isaac Gaon in Baghdad
Baghdad – In a vibrant Baghdad district, labourers are working tirelessly to repair the centuries-old shrine of a revered rabbi in an effort to revive the long-faded heritage of Iraq's Jewish community. A few months ago, the tomb of Rabbi Isaac Gaon was filled with rubbish. Its door was rusted, the windows shattered and the walls stained black from decades of neglect. Today, marble tiling covers the once-small grave, and at its centre stands a large tombstone inscribed with a verse, the rabbi's name and the year he died: 688. A silver menorah hangs on the wall behind it. 'It was a garbage dump and we were not allowed to restore it,' said the head of Iraq's Jewish community, Khalida Elyahu, 62. The Jewish community in Iraq was once one of the largest in the Middle East, but now it has dwindled to just dozens. Baghdad today has one synagogue left, but it has no rabbis. And many houses that once belonged to Jews are abandoned and dilapidated. The Jewish community itself is funding the shrine's restoration, at an estimated cost of $150,000. The project will bring 'a revival for our community, both within and outside Iraq', Elyahu said. With the backing of Iraqi officials, she said she hopes to restore more neglected sites. Little information is available about Rabbi Isaac. But when Iraq's National Security Adviser Qassem al-Araji visited the tomb earlier this year, he said the rabbi had been a finance official. Baghdad today has one synagogue left, but it has no rabbis Rabbi Isaac Gaon was prominent during the Gaonic period, also known as the era of Babylonian academies for rabbis. The term 'Gaon' is likely to refer to his position as the head of one such academy. His name was mentioned in the 10th century by another rabbi, who told a tale that never appeared elsewhere, according to Professor Simcha Gross from the University of Pennsylvania. 'There is only one single story,' said Gross. It goes that Rabbi Isaac led 90,000 Jews to meet Ali Ibn Abi Talib, the fourth Islamic caliph and a relation of the Prophet Mohammed, who is also revered by Shiites as the first Imam, during one of his conquests in central Iraq. 'We have no other evidence for this event, and there are reasons to be sceptical,' Gross said. Nothing else is known about Rabbi Isaac, not even his religious opinions. But the tale has origins that are not without context, said Gross. In the 10th century, minorities — Christians, Jews, and Zoroastrians — began to tell stories of how they greeted 'Muslim conquerors' because 'their privileges including taxes were dependent on whether or not they were believed to have welcomed the Muslims', he said. At that same time, Jewish shrines started to appear, even though Jewish roots in Iraq date back some 2,600 years. According to biblical tradition, Jews arrived in Iraq in 586 BC as prisoners of Babylonian king Nebuchadnezzar II after he destroyed Solomon's Temple in Jerusalem. In Iraq, they wrote the Babylonian Talmud. Thousands of years later, in Ottoman-ruled Baghdad, Jews made up 40 percent of the population. A turning point was the 1941 pogrom in Baghdad when more than 100 Jews were killed. Like other Jewish communities in the Arab region, their history has changed since the Palestinian Nakba — 'catastrophe' in Arabic — and Israel's creation in 1948. Soon afterwards, almost all of Iraq's 135,000 Jews went into exile. Decades of conflict and instability — Saddam Hussein's dictatorship, the 2003 US-led invasion and the ensuing violence — completed the community's erosion. The Jewish community in Iraq was once one of the largest in the Middle East, but now it has dwindled to just dozens Some who stayed on converted to other religions, or do not reveal their faith. Today, 50 synagogues and Jewish sites remain, Elyahu said. Most are crumbling, and some have become warehouses. Rabbi Isaac's shrine once included a synagogue and a school, but has been reduced to the small room housing the grave, the restoration's supervisor said. 'It took us two months to clean it of garbage,' said the supervisor, who asked to remain anonymous. Now 'we are receiving requests from outside Iraq to visit it'. Decades ago people would come to pray and light candles, believing in the rabbi's 'healing powers'. Mussa Hayawi, 64, lives nearby. He recounted stories from his childhood in a quarter which was, until the 1940s, one of several Jewish districts in Baghdad. He said women used to soak themselves in water from the shrine's well, hoping to conceive. Rabbi Isaac 'was a revered man'. People came 'to pray for their sick, to ask for a baby, or the release of a prisoner', Hayawi said.
Time of India
5 days ago
- Politics
- Time of India
Iraq's Jewish community saves a long-forgotten shrine
Representative AI image BAGHDAD: In a vibrant Baghdad district, labourers are working tirelessly to repair the centuries-old shrine of a revered rabbi in an effort to revive the long-faded heritage of Iraq's Jewish community. A few months ago, the tomb of Rabbi Isaac Gaon was filled with rubbish. Its door was rusted, the windows shattered and the walls stained black from decades of neglect. Today, marble tiling covers the once-small grave, and at its centre stands a large tombstone inscribed with a verse, the rabbi's name and the year he died: 688. A silver menorah hangs on the wall behind it. "It was a garbage dump and we were not allowed to restore it," said the head of Iraq's Jewish community, Khalida Elyahu, 62. The Jewish community in Iraq was once one of the largest in the Middle East, but now it has dwindled to just dozens. Baghdad today has one synagogue left, but it has no rabbis. And many houses that once belonged to Jews are abandoned and dilapidated. The Jewish community itself is funding the shrine's restoration, at an estimated cost of $150,000. The project will bring "a revival for our community, both within and outside Iraq", Elyahu said. With the backing of Iraqi officials, she said she hopes to restore more neglected sites. Who was he? Little information is available about Rabbi Isaac. But when Iraq's National Security Adviser Qassem al-Araji visited the tomb earlier this year, he said the rabbi had been a finance official. Rabbi Isaac Gaon was prominent during the Gaonic period, also known as the era of Babylonian academies for rabbis. The term "Gaon" is likely to refer to his position as the head of one such academy. His name was mentioned in the 10th century by another rabbi, who told a tale that never appeared elsewhere, according to Professor Simcha Gross from the University of Pennsylvania. "There is only one single story," said Gross. It goes that Rabbi Isaac led 90,000 Jews to meet Ali Ibn Abi Talib, the fourth Islamic caliph and a relation of the Prophet Mohammed, who is also revered by Shiites as the first Imam, during one of his conquests in central Iraq. "We have no other evidence for this event, and there are reasons to be sceptical," Gross said. Nothing else is known about Rabbi Isaac, not even his religious opinions. But the tale has origins that are not without context, said Gross. In the 10th century, minorities, Christians, Jews, and Zoroastrians, began to tell stories of how they greeted "Muslim conquerors" because "their privileges including taxes were dependent on whether or not they were believed to have welcomed the Muslims", he said. At that same time, Jewish shrines started to appear, even though Jewish roots in Iraq date back some 2,600 years. According to biblical tradition, Jews arrived in Iraq in 586 BC as prisoners of Babylonian king Nebuchadnezzar II after he destroyed Solomon's Temple in Jerusalem. In Iraq, they wrote the Babylonian Talmud. Thousands of years later, in Ottoman-ruled Baghdad, Jews made up 40 percent of the population. A turning point was the 1941 pogrom in Baghdad when more than 100 Jews were killed. Like other Jewish communities in the Arab region, their history has changed since the Palestinian Nakba -- "catastrophe" in Arabic -- and Israel's creation in 1948. Soon afterwards, almost all of Iraq's 135,000 Jews went into exile. Blessings Decades of conflict and instability, Saddam Hussein's dictatorship, the 2003 US-led invasion and the ensuing violence completed the community's erosion. Some who stayed on converted to other religions, or do not reveal their faith. Today, 50 synagogues and Jewish sites remain, Elyahu said. Most are crumbling, and some have become warehouses. Rabbi Isaac's shrine once included a synagogue and a school, but has been reduced to the small room housing the grave, the restoration's supervisor said. "It took us two months to clean it of garbage," said the supervisor, who asked to remain anonymous. Now "we are receiving requests from outside Iraq to visit it". Decades ago people would come to pray and light candles, believing in the rabbi's "healing powers". Mussa Hayawi, 64, lives nearby. He recounted stories from his childhood in a quarter which was, until the 1940s, one of several Jewish districts in Baghdad. He said women used to soak themselves in water from the shrine's well, hoping to conceive. Rabbi Isaac "was a revered man". People came "to pray for their sick, to ask for a baby, or the release of a prisoner", Hayawi said.
