In medieval Europe, Italians regarded 6 as a number belonging to the goddess of love Venus, believing it to be a symbol of a happy marriage. It was also known as a perfect number. In fact, when it was introduced to China, perfect numbers had another translation, called "perfect numbers". If the sum of all the factors (excluding itself) of a natural number is exactly equal to that number, then such a natural number is called a perfect number. It can be seen that 6 has three factors 1, 2, and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number and the smallest member of the perfect number family. The number 28 has five factors 1, 2, 4, 7, and 14, and it is the second perfect number: 1 + 2 + 4 + 7 + 14 = 28. All the discovered perfect numbers are even numbers, not a single odd number. Despite this, people still haven't been able to prove that odd perfect numbers don't exist, and this problem has become another unsolved puzzle in number theory.
As early as more than 500 years before Christ, Pythagoras had already come into contact with perfect numbers. Later, Euclid discussed perfect numbers in Chapter 9 of his "Elements of Geometry", pointing out that if 2^n - 1 is a prime number, then 2^(n-1)(2^n - 1) is a perfect number. By the 17th century, French Catholic monk Mersenne had done a lot of calculations on 2^n - 1 and made some assertions (some of which were wrong). In memory of his achievements, later generations called numbers of the form 2^n - 1 Mersenne numbers and recorded them as Mn.
It can be easily proven that when a Mersenne number 2^n - 1 is a prime number, n must be a prime number; conversely, when n is a prime number, the Mersenne number seems to be a prime number, for example: M3 = 7, M5 = 31, M7 = 127. However, unexpectedly, when the next M11 = 2^11 - 1 = 2047 = 23×89 appeared, this rule no longer existed! Due to the many peculiar properties and fascinating anecdotes of Mersenne primes, they have attracted many mathematicians for centuries. However, until the 18th century, only eight Mersenne primes were discovered, namely M2, M3, M5, M7, M13, M17, M19, and M31. Among them, the last M31 was proved by mathematician Euler by mental calculation when he was blind in both eyes, which was the largest known prime number in the world at that time. Euler also proved the converse of Euclid's theorem on perfect numbers, that is, every even perfect number has the form 2^(n-1)(2^n - 1), where 2^n - 1 is a prime number. This made even perfect numbers a "by-product" of Mersenne primes. In the following 100 years, only four new Mersenne primes were discovered - M61, M89, M107, and M127. Among them, the important theorem proposed by French mathematician Lucas in 1876 to determine whether Mp is a prime number - Lucas's theorem - provided a powerful tool for the search for Mersenne primes. In the "era of manual calculation and paper recording", people struggled to find only 12 Mersenne primes.
The advent of electronic computers greatly accelerated the search for Mersenne primes. In 1952, mathematicians Robinson and others compiled the Lucas method improved by American mathematician Lame in 1930 into a computer program and used the SWAC computer to find five new Mersenne primes: M521, M607, M1279, M2203, and M2281. In 1963, when the 23rd Mersenne prime number M11213 was found by a large computer, the American Broadcasting Company interrupted its normal program broadcast and released this important news in the first time. All the faculty and students of the Department of Mathematics of the University of Illinois in the United States who discovered this prime number felt extremely proud. In order to let the whole world share this achievement, they stamped "2^11213 - 1 i" on all the letters sent from the department.The postmark of "prime" (2^11213 - 1 is a prime number). It is worth mentioning that two American middle school students, Noll and Nicole, who were just starting out, spent three years writing a calculation program and discovered the 25th Mersenne prime number M21701 on the Cyber174 computer in October 1978. At that time, major news agencies around the world (including Xinhua News Agency of China) and academic journals competed to report this news, and The New York Times even reported it as the headline of the front page.
The emergence of the new technology of grid computing has greatly enhanced the search for Mersenne primes. In early 1996, American mathematician and programmer Wolfram developed a Mersenne prime number calculation program and put it on the web for mathematicians and math enthusiasts to use for free, which is the world-famous "Internet Mersenne Prime Search" (GIMPS) project. The project adopts grid computing, using the idle time of a large number of ordinary computers to obtain the computing power equivalent to that of supercomputers. Now, as long as people go to the GIMPS homepage to download a free program, they can immediately participate in the GIMPS project to search for Mersenne primes. At present, nearly 150,000 people from more than 150 countries and regions around the world participate in the international cooperation project of GIMPS. People have found 14 new Mersenne primes through the GIMPS project. On August 23, 2008, computer expert Smith of the University of California, Los Angeles discovered the 45th Mersenne prime number M43112609. On September 6, 2008, electronic engineer Elvinico discovered the 46th Mersenne prime number M37156667. On April 12, 2009, Norwegian Strindmo discovered the Mersenne prime number M42643801 between the above two numbers. On January 25, 2013, a research team led by Curtis Cooper, a mathematics professor at the University of Central Missouri, discovered the largest known Mersenne prime number M57885161, which has more than 17 million digits, bringing the total number of known Mersenne primes to 48. Except for the first one, all Mersenne primes end with 1 and 7. Interestingly, most of the 48 Mersenne primes end with 1.
Why are people so keen on finding Mersenne primes? From the ancient Greek era to the 17th century, the significance of searching for Mersenne primes seemed to be just to find perfect numbers. But since Mersenne put forward his famous conjecture, especially since Euler proved the converse of Euclid's theorem on even perfect numbers, perfect numbers have only been a "by-product" of Mersenne primes. Searching for Mersenne primes is the most effective way to find the largest known prime numbers. Since Euler proved that M31 was the largest prime number at that time, Mersenne primes have almost won all the championships in the world-wide competition for finding the largest known prime numbers. Searching for Mersenne primes is also a powerful means of testing computer computing speed and other functions. For example, the 34th Mersenne prime number M1257787 was obtained by the US company Cray in September 1996 when it was testing the computing speed of its latest supercomputer.
Mersenne primes also have practical applications. Nowadays, people have used large prime numbers in the field of modern cryptography design. The principle is that it is very difficult to factorize a large number into the product of several prime numbers, but it is much easier to multiply several prime numbers together. In this kind of cryptography design, large prime numbers are needed, and the larger the prime number, the lower the possibility of the password being cracked.