April 6, 2000, Mr. Nayan Hajratwala, who lives in the US state of Moutz, Michigan, got a $50,000 math bonus because he found the largest prime number known to date, which is a mason prime:

26972593-1.

This is also the first number of digits we know to be more than 1 million digits. To be precise, if you write this prime number in our familiar decimal form, it has a total of 2,098,960 digits, and if it is written in this form, it takes about 150 to 200 pages.

But Mr Hagiratvala is not a mathematician, and he may even be ignorant of the mathematical theory of finding prime numbers-though this earned him the prize. All he did was download a program from the Internet. This program runs silently when he does not use his Pentium II350 computer. After 111 days of calculation, the prime number mentioned above was found.

Second, Mason Prime number

We have a natural number greater than 1 called Prime, if only 1 and it itself can divide it. If a natural number greater than 1 is not prime, we call it composite. 1 is neither prime nor composite.

For example, you can easily verify that 7 is a prime number, and 15 is a composite number, because 3 and 5 can be divisible by 15 except 1 and 15. By definition, 2 is a prime number, and it is the only number of even primes. As early as the Three century ago in the ancient Greek era, the great mathematician Euclid proved that there are infinite number of primes.

There are many simple and beautiful, but extremely difficult, questions about prime numbers that have not been answered yet. One of the famous Goldbach conjectures is that any even number greater than 6 can be expressed as the sum of two odd primes. There is also the question of twin primes. The prime pairs, which differ by 2, like 5 and 7,41 and 43, are called twin primes. The twin prime question is: Is there infinitely many pairs of twin primes? Here, incidentally, these seemingly simple mathematical problems, their solutions will certainly be extremely complex and require the most advanced mathematical tools. If you are not arrogant enough to think that hundreds of or even thousands of of years have spent countless talents on these issues (many are very great) and math enthusiasts are not as smart as you are, do not try to use the elementary method to solve these problems, time-consuming and energy.

The ancient Greeks were also interested in another number. They call it the perfect number. A natural number greater than 1 is called the perfect number, if all of its factors (including 1, but not including itself) are equal to itself. For example, the 6=1+2+3 is the smallest perfect number, the ancient Greeks regarded it as Venus is the symbol of love. 28=1+2+4+7+14 is another perfect number. Euclid proved that an even number is perfect when and only if it has the following form:

2p-1 (2p-1)

Where 2p-1 is prime. The 6 and 28 above correspond to the case of p=2 and 3. All we have to do is find a prime that is shaped like a 2p-1, and we know a perfect number, and we find all the perfect numbers just by finding all the primes that are like 2p-1. So Mr. Hagiratvala not only found the largest primes known in the world, but also found the largest number of even perfect numbers known in the world. Well, you ask, what about the odd perfect number? The answer is: we have not even found a perfect number, we do not know whether there is a singular perfect number exists. We only know that if there is a singular perfect number, it must be very, very big! The question of whether the odd perfect number exists is also a well-known mathematical problem, which is simple and beautiful, but extremely difficult.

For a long time people thought for all prime numbers p,

M_p=2p-1

Are primes (note that to make 2p-1 a prime, p itself must be a prime, think about why?) But in 1536 Reguios (Hudalricus regius) pointed out that m_11=211-1=2047=23*89 is not prime.

Pitro Catardi (Pietro Cataldi) Firstly, this kind of number is studied systematically. In his 1603 announcement, he said that for p=17,19,23,29,31 and 37,2p-1 are primes. But the 1640 Fermat used the famous Ferma theorem (not to be confused with that Ferma theorem) to prove that Cataldi's results on p=23 and 37 were wrong, and Euler proved in 1738 that the result of p=29 was also wrong, and later he proved that the conclusion about p=31 was correct. It is worth pointing out that Cataldi was using a manual one to obtain his conclusions, while Fermat and Euler used the most advanced mathematical knowledge at their time, avoiding many complex computations and thus possible errors.

The French priest, Mason (Marin Mersenne), published his results in 1644. He claims that for both p=2,3,5,7,13,17,19,31,67,127 and 257,2p-1 are primes, and for other primes less than 257, p,2p-1 are composite numbers. Today we call the prime number of m_p=2p-1, the number of primes, and the M in M_p is the first letter of Mason's surname.

It is quite difficult to judge a large number by hand, and father Mason himself admits that his calculations are not necessarily accurate. Until a century later, in 1750, Euler announced the finding of Father Mason's mistake: m_41 and m_47 are also primes. But greatness, like Euler, also makes computational mistakes--in fact m_41 and m_47 are not primes. But that's not to say that Father Mason's results are right. Until 1883, when Father Mason's results were announced for more than 200 years, the first error was discovered: M_61 is a prime. The other four errors were also found: m_67 and m_257 are not primes, and m_89 and m_107 are primes. It was not until 1947 that the correct result of the m_p of P<=257 's Mason primes was determined, that is, when p=2,3,5,7,13,17,19,31,61,89,107 and 127, m_p were primes. Now that the table has been repeatedly validated, there must be no mistakes.

Here is a list of all the Mason primes we know now: (We notice that Father Mason's name is not on it-the primes have been named after him, giving the honors to the last confirmed.) ）

Is there an infinite number of Mason primes? Mathematicians are not yet able to answer this question.

Three, looking for a larger prime

Why do you want to find a mason prime? Why break the record of known maximum primes? What's the use of that?

If what you are saying is that you can create material wealth directly, then I have to tell you that there is no use for Mason primes, and that it seems to be of no use to know a very large number of primes. Even if we knew a giant Mason Prime, it wouldn't add a penny to our wallet (hey, wait!) If you are only interested in money, please do not leave my article immediately. I mean, what I said above was to rule out the 100,000-dollar bonus I mentioned in the title of this article--your wallet might be bulging. So be patient, please.

But human beings do not just need material wealth. What's the use of the Diamonds in the museum? Why do humans collect them? Because they are beautiful and scarce. As a crystallization of human wisdom, Prime, Mason primes and the perfect number closely related to it are very beautiful. They are simple in definition, yet so mysterious and unpredictable, great mathematicians such as Euclid, Descartes, Fermat, Leibniz, and Euler have done a great deal of research on it because of their beauty, and we have seen that for more than 2000 of years, after countless generations of hard work, we have collected only 38 primes, They are very rare. For mathematicians, collecting primes, Mason primes and perfect numbers is as fun as collecting diamonds.

Man also needs glory--perhaps more than wealth. In sports, to be able to run faster, jump a bit higher, is there really a practical material use? No, we like to accept the challenge, we hope to win. Breaking a sports world record, climbing Mount Everest, sailing across the Pacific Ocean ... that is a challenge to the limits of human physical fitness, and looking for larger primes is a challenge to human intelligence. When we have done an unprecedented task, we always feel extremely proud. When the 23rd Mason was found in 1963, the University of Illinois Mathematics Department was so proud that all the letters from the department were stamped with the "211213-1 is a prime" postmark.

After Euler proved that m_31 was prime, the next largest prime number was recorded by Landry (Landry) in 1867: m_59/179951=3203431780337. This is not a mason prime. This record was maintained for nine years.

The 1876 Edward Lucas used a more advanced method than the Fermat and Euler methods, proving that m_127 is a prime. This record was maintained for 75 years. Until Ferrier (Ferrier) used a hand-cranked computer in 1951 to prove that (2148+1)/17 was a prime number, and it had 41 digits.

The use of hand-cranked computer methods to be counted as manual calculation method or to be counted as a computer method, is probably a problem to explore. But the development of technology all of a sudden makes this argument unnecessary. It is worth pointing out that the improvement of mathematical theory is far more important than the ability to compute with strong tenacity in the search for large prime numbers in humans. Lucas's approach was simplified by Flamel (Lehmer) in 1930, and the Lucas-Flamel test is now the standard way to look for Mason primes.

(Lucas-Flamel test: For all odd-numbered p,m_p greater than 1 are primes and only if m_p is divisible by S (p-1), where S (n) is defined by the n+1 of S (=s) 2-2,s (n) =4 (1). This test is especially suitable for computer operations, since dividing the m_p=2p-1 operation in binary can be accomplished simply by using the shift and addition operations that the computer is particularly adept at. The way to judge a mason number as a prime is much simpler than a method of determining the number of other types that are about the same size, so most records are in the process of finding the largest primes. ）

In 1951, Miller and Edsac (Miller & Wheeler) found a 79-bit prime 180 (m_127) 2+1 with the help of the computer, which is less than the general calculator we now use, with only 5 K of memory. The record has not been able to hold for long. The following year Robinson applied the SWAC computer and discovered the 13th and 14th Mason primes in early 1952: m_521 and m_607, followed by three consecutive Mason primes: m_1279,m_2203 and m_2281.

In the years that followed, computers used to break the record of huge primes were becoming more powerful, including the famous IBM360-type computer, and the Cray series of supercomputers. You can see this competition process by reference to the above list of Mason primes. In the meantime, there is only one time that a prime not of Mason sits on the throne of the "known largest prime", which is 39158*2216193-1, which was discovered in 1989. m_1257787, discovered in 1996, was the last mason to be discovered by supercomputers, and mathematicians used Cray T94.

Then came the time of Gimps.

Four, gimps--Internet of Mason Prime number search

1995 Program Designer George Waterman (George Woltman) began collecting data on the calculation of the Mason prime number. He compiled a mason prime number search program and placed it on the web for free use by math enthusiasts. This is the "Internet Mersenne Prime Number Search" program (gimps,the great internet Prime Search). In this program, more than 10 math experts and thousands of math enthusiasts are looking for the next largest number of Mason, and examining the previously unexplored gaps between the records of the Mason primes. For example, in the above list of the Mason Prime, the last number of the primes is unknown, we do not know the number 37th and whether there are other undiscovered Mason primes.

1997 Scott Kurwoski (Scott Kurowski) and others established the "Prime Network" (primenet), automating the allocation of search intervals and sending reports to Gimps. Now as soon as you go to Gimps's homepage to download that free program, you can immediately join the GIMPS program to search for Mason primes. Almost all of the common computer platforms have versions available. Programs run on your computer with the lowest priority, so it has little impact on your normal use of your computer. The program can also be stopped at any time, and it will continue to compute from where it was stopped the next time it starts.

From 1996 to 1998, the Gimps plan found three primes: m_1398269, m_2976221, and m_3021377, all of which were the result of using a Pentium-type computer.

In March 1999, an association "electronic Border Fund" (Eff,electronic Frontier Foundation), which was active on the Internet, announced a bonus for the search for large prime numbers, funded by an anonymous person. It provides a 50,000 dollar bonus to the first person or institution to find a prime number of more than 1 million digits, which is the prize we Hagiratvala to get. The following bonuses were in turn: More than 10 million, 100,000 dollars, more than 100 million, 150,000 dollars, more than 1 billion, 250,000 dollars.

The verification of search results and the issuance of bonuses are very stringent. For example, the resulting result must be explicit-you cannot claim that your result is a solution of a set of 100 equations without solving it. The result must be verified independently by another computer. All of these rules are explained on the EFF website.

It should be noted that the hope of getting bonuses by participating in the GIMPS program is rather small. Hagiratva used a computer that was one of 21000 computers at the time. Each participant was validating the different Mason numbers assigned to him, and of course most of them were not prime--he had only about a one out of 10,000 chance of encountering a prime.

The next 100,000 dollar bonus will be awarded to the first person or institution to find a prime number of more than 10 million digits. This time the calculation will be approximately equivalent to 125 times times the previous one. Gimps now has the ability to compute 700 billion floating-point operations per second, and one of today's most advanced hyper-vector computers, such as Cray T932, is quite capable of running. But if Gimps were to use such supercomputers, they would have to pay about 200,000 dollars a day. And now they need the cost of supporting the operation of the Web site, and a total of hundreds of thousands of dollars in bonuses.

V. Online Distributed computing Plan

Gimps is just one of the many distributed computing programs on the Internet, and the Gimps home page has an introduction to these plans.

Distributed computing is a computer discipline that studies how to divide a problem that requires a very large computational capability into a number of small parts, then allocate these parts to many computers for processing, and finally combine these calculations to get the final results. Sometimes the computation is so large that it takes thousands or even more computers around the world to work together to get results in a reasonable time. The Gimps program is doing this kind of distributed computing.

But it is not the most famous distributed computing program. The SETI@home project in the Search for Extraterrestrial Civilizations (SETI program), which is dedicated to finding intelligent life in the universe, has recruited 2.9 million people worldwide (!) Volunteers, using a screen saver to deal with a large number of cosmic radio signals received by radio telescopes. If you take part in this program, you may someday decipher the aliens ' greetings on your computer.

You can also use your computer's spare computing power to make a contribution to human conquest of cancer. British scientists have designed a distributed computing screensaver similar to the SETI@home project, which downloads data from the Web site, analyzes the anticancer properties of chemical molecules, and then passes the results back to researchers as a reference for developing new anticancer drugs. The project will be officially launched on April 3, 2001 in the State of California, USA.

The updating of computer hardware is dizzying, and the latest PC, bought in the first half of the year, became Da Lu Huo in the second half. The CPUs of three or four years ago are now worthless--perhaps not, you can't buy them--the cheapest CPUs on the market are much more powerful than they are. and an ordinary home computer running for five consecutive years is no problem. So the most economical way to treat a computer is to keep it running.

And there are so many things to calculate, there are so many problems to find answers, there are so many difficulties to overcome. We need more and more computing power, and we have the ability to compute, but too much is wasted away in vain. The internet has made it possible to make large-scale distributed computing plans. Now, the only thing we need is the willingness and confidence of computer users at every node of the web.

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