Challenging Perfect Number

Passage and pictures by Huang Hongtao who is the post-doctorate from Università di Roma La Sapienza and does researches on computational mathematics

On January 7 this year， a mathematician named Cooper from the University of Central Missouri Cooper， through the project called Internet Mersenne Prime Search （GIMPS）， found the largest known perfect number 2 ^ 74，207，280 （2 ^ 74207281-1） which means that after 74，207，281th power of 2 minus 1 and then multiplied by 2 to the power of 74，207，280 （NOTE： ^ is the computer language power symbol）. This number is the 49th perfect number human discovered during the past 2500 years and it has 44，677，235 digits； if it were printed in the ordinary size， the length will be 200 km！ Readers can imagine more about it ...... Even the well-known Australian mathematician Parker agrees that this is a tremendous scientific achievement. Well， will it be an explorative story similar to the classic movie Good Will Hunting？ What kind of charm on earth does the perfect number boast， attracting so many mathematicians on after another to devote themselves？

What is Perfect Number？

Perfect number has some other names in Chinese whose definition is that the sum of all factors except itself equals the number itself. This definition seems to be confusing， so lets cite two examples， the two smallest perfect numbers are 6 whose all factors are 1，2，3，6 and 28 whose all factors are 1，2，4，7，14，28）， both of which are the sum of their all factors in addition to itself： 6 = 1 + 2 + 3 and 28 + 1 = 2 + 4 + 7 + 14. Due to some common magical properties， scientists entitle them with a wonderful name called the perfect number.

For thousand years， the intriguing properties and incomparable charm of the perfect number has attracted many mathematicians and countless mathematic amateurs to explore it. In 17th century， French mathematician and philosopher Descartes publicly predicted， “The number of the perfect number will not be large， just like its not easy to find a perfect person. " After a long period of time， people only find 49 perfect numbers. This number of rare yet beautiful which is known as the "diamonds in treasure house of number theory."

Discovery of Mersenne Prime

Later， the structure of 2P-1 using for finding the perfect number was named as Mersenne Prime in the mathematical field. It was named after Mersenne who is a French mathematician because his systematic and deep research into this special prime. Interestingly， the "super primes" found in the past century are almost Mersenne primes.

Mersenne prime seems to be simple but actually its hard to explore. It requires not only profound theories and skillful techniques， but also complicated calculations and great amount of computation. In 1772 when Euler was blind， he still proved 231-1 （ie 2147483647） to be the eighth Mersenne prime by mental arithmetic. This prime with 10 digits was the largest known prime number at that time. The eighth perfect number --230 （231-1） also emerged which was likely to be the biggest perfect number people found at that moment. Euler's perseverance and problem-solving skills were impressive. What French mathematician Laplace says maybe can represent our voice： "Read Eulers writings. He is the teacher of everyone of us."

Against the backdrop where all the calculations and record must be written down by hands， no matter how hard people tried， they found only 12 Mersenne primes， that is to say， only 12 perfect numbers were discovered. However， the birth of computers greatly accelerated the process of research into Mersenne primes. For example， in 1952， the American mathematician Robinson compiled "Lucas - Lehmer test" into a computer program. By using the SWAC computer in a few months， they found five Mersenne primes： 2521-1，2607 -1，21279-1，22203-1 and 22281-1.

Exploration into Mersenne prime is not only challenging. For the explorers， they can obtain a sense of pride. Perhaps this is why there are countless mathematicians willing to devote themselves into it. At 20：00 on June 2， 1963， when the first 23rd Mersenne prime 211213-1 was found by large computer， the American Broadcasting Corporation （ABC） interrupted its regular broadcast and timely published this important message. All the students and faculties from the Department of Mathematics， University of Illinois staff who found the prime were very proud and at the same time， in order to let the world to share this great achievement， all the envelopes sent from the department are covered with a "211213-1 is a prime number "postmark， which was very crazy and fantastic.

With the increase of the index P， the discovery of every Mersenne prime encounters tremendous hardships. However， the professional mathematicians and amateur mathematics lovers were not tired of this and even tried to take the lead. On February 23， 1979， when Vinski and Nelson who are the computer experts from the American Cray Research Company computer announced that they have found the first 26th Mersenne prime 223209-1， someone told them that just two weeks ago， a high school student named Noel had given the same result. Therefore， they devoted themselves strenuously， and spent one and a half months， finding the new Mersenne prime 244497-1 by using the Cray-1 computer. This event hit headlines of many mainstream newspapers at that time. Later， Los Wenski found six Mersenne primes by himself and was hailed as "king of prime numbers."

What is worth mentioning is that while people are looking for Mersenne prime， they also research into the rule of its distribution. According to the known Mersenne primes， their distribution in the positive integers are sometimes sparse and sometimes dense which is extremely irregular， therefore， studying on the distribution of Mersenne primes seems to be more difficult than finding a new Mersenne prime.

Mathematicians from Britain， France， Germany and the United States have all predicted about the distribution of Mersenne primes which are almost approximate expressions and are not that close to the actual situations. On February， 1992， Zhou Haizhong， a Chinese mathematician and linguist， first gave out the precious expression of the distribution of Mersenne prime after years efforts. Later， this major achievement was named "Zhous Guess" internationally. Vekselberg who is the Norwegian-American master of number theory and the winner of the Fields Medal and Wolf Prize believes Zhous guess is innovative which creates a new inspiring method and its innovative nature is also reflected in the reveal of new rules .

Later， the emergence of distributed computing technology facilitated on the exploration of the Mersenne prime. At the beginning of 1996， Waterman， a US computer expert， compiled a calculation program for the Mersenne prime and upload it on the Internet for both professional mathematicians and amateur mathematics lovers for free. This is the world-famous GIMPS project， and it is also the world's first internet-based distributed computing project. The project mainly uses the unused processing capacity of a large number of ordinary computers get the computing power which can be compared with the super computer. The US computer expert Kurt Voss in 1997 established the "prime network" to automate the search of domain distribution and the process of sending reports to GIMPS. As long as we download the the project's Prime95 or MPrime software from the open source， we can immediately find the Mersenne primes.

In order to encourage people to find Mersenne primes and promote the development of grid technology， the Electronic Frontier Foundation （EFF） headquartered in the United States announced to the world to establish Collaborative Computing Award in March 1999， aiming at finding Mersenne primes through GIMPS project.

In June 1999， the mathematic lover， Nayan Hajratwala who lived in Michigan found the first prime with more than one million digits， that is 26972593-1 through GIMPS project and became the first one to get the prize money. This number is the 38th prime number and the last one in the 20th century. NayanHajratwala was the CEO of a company. He said when he received the interview， “ Two years ago with curiosity and thirst for knowledge， I took part in the GIMPS project， and Im lucky enough to find the treasure only after three weeks continuous operation on computer. "

Smith， a computer expert from the University of California， Los Angeles first discovered in 2008 the Mersenne prime with more than 10 million digits --243112609-1， or rather， this number has 12，978，189 digits and therefore he also won the prize money worthing 100，000 awarded by EFF. This major achievement was named as one of the “50 Best Inventions of 2008" by Times. However， Smith attended the GIMPS project without getting permission of using 75 computers in the university， which should get punished. However， since he won the glory for the school， he received compliments instead.

At present， there are 192 countries and regions and more than 60 million people in the world using more than 1.25 million central processing unit （CPU） to participate into GIMPS project. So far， 15 Mersenne primes have been found through this project； it can be said that people through the project has found 15 perfect number. Those who discovered the perfect numbers come from the United States， Germany， Britain， France and other places. Sautoy， the chairman of the British Association of Mathematics and the author of the book named Music of Primes， said when he gave a lecture in Naning in 2009 that he hopes to see Chinese people join the discovery of Mersenne prime.

Application Trends of Mersenne Prime

The family of perfect number reflects some basic law of the natural numbers and now its practical use is under constant exploration. However， the value of Mersenne prime which is the key part of the perfect number is gaining more and more attention in the contemporary society.

With regard to computer detection technology， searching forMersenne primes can find problems in computer chips. Recently， a German participant of GIMPS project found that when using Intel's sixth-generation core processor Intel Skylakeand execute Prime95 application to find Mersenne primes， the system crashes if the operation comes to the index P = 14942209. In fact， from the beginning of the 1990s， the Cray Company and Apple of United States began to use Mersenne primes to test the functionality of the computer whose principle is tocontinue the operation ofMersenne primes by CPU so that the CPU is operating under heavy load which is also a chance to test the stability of their systems. In addition， the Mersenne primes in cryptography has potential applications， that is， in cryptology， you need to use a large prime number， and the greater the prime number is， the less likely the password is deciphered.

Finally， what is worth mentioning is that the 49 perfect numbers indeed are all even. So， if there is an odd perfect number？ In addition， the number of perfect numbers is unlimited？ These problems are well-known mathematical problem which looking forward to be solved by later generations.