1. Value of 50,000 US dollars on April 6, 2000, live in Plymmaoz, Michigan, USA, got a 50,000 dollar math bonus, Mr. Hajratwala. Because he found the maximum number of people known so far, this is a number of Mason: 26972593-1. This is also the number of prime numbers that we know more than one million. Accurately, if this number is written to our familiar decimal form, it has two hundred and nine thousand nine hundred and sixty-six numbers. If it writes it in this form, it takes approximately 150 to 200 articles. . However, Mr. Hajaratwara is not a mathematician. He is even likely to know anything about the mathematics theory of finding the number of mathematics - although this makes him win this bonus. Everything he did is to download a program from the Internet. This program runs quietly when he doesn't use his Pentium II350 computer. After 111 days of calculation, the number of this prime mentioned above was discovered. Second, Messen's number we put a number of natural numbers greater than 1, if only 1 and it itself can be eliminated. If a large number of natural natural numbers is not a prime number, we call it to them. 1 is neither a number of prime, nor is it a compliment. For example, you can easily verify that 7 is a number of prime; and 15 is a compliment, because in addition to 1 and 15, 3 and 5 can be removed 15. Depending on the definition, 2 is a prime number, it is the only counted number. As early as three hundred years ago, the great mathematician Ou Sri proved that there is an infinite number of pins. With regard to the number of prime, there are many simple and beautiful, but it is extremely difficult, and there is no answer to the answer now. Among them, there are famous Gothbach guesses that it is said that any even number of more than 6 can be expressed as the sum of the number of two qi. There are also twin coupons. The number of pskits 2 of 5 and 7, 41 and 43 is called twin of twins. The twin number of twins is said: Is there an infrequent number of twins? Here is that these seem simple mathematics issues, their solutions will be extremely complex and requires the most advanced mathematical tools. If you are not arrogant to think that all the mathematicians of many or even thousands of mathematicians (many are very great) and mathematics enthusiasts are not as smart, do not try to use the first, etc. Methods to solve these problems, spending time and energy. Ancient Greeks are also interested in another number. They call it a perfect number. A natural number of large than 1 is called perfect, if it's all factors (including 1, but not included), it is equal to it itself. For example, 6 = 1 2 3 is the smallest perfect number, ancient Greeks see it as a symbol of love. 28 = 1 2 4 7 14 is another perfect number. The European Milled proved that an even number is perfect, when it has the following form: 2P-1 (2p-1) where 2P-1 is the number of prime. The upper 6 and 28 correspond to the case of P = 2 and 3. As long as we find a number of pskins such as 2P-1, we also know a number of elders; we only find all the possibility of 2P-1, and find all the satisfaction. Therefore, Mr. Hajarat Wara not only found the largest number of people known in the world, but also found the biggest even satisfaction in the world. Well, you have to ask, what is the situation about the fare? The answer is: We haven't found a grateful number now. We don't even know if there is a singularity. We only know that if there is a quicker, it must be very very big! Whether there is such a problem with the american number, it is also a simple and beautiful mathematics issue that is just simple and beautiful, but very difficult.
For a long time, people think that the number of numbers of all prime numbers (note that 2P-1 is a prime number, P itself must be a number of prime, thinking about why?) But in 1536 Hudalricus Regius pointed out that m_11 = 211-1 = 2047 = 23 * 89 is not a number of prime numbers. Pietro Cataldi first conducted a systematic study of this type. He said in the results announced in 1603 that for P = 17, 19, 23, 29, 31, and 37, 2P-1 is the number of prime. However, 1640 Fei Ma was using the famous Felmmine minimal (not confused with the felmada) to prove that Qatardi's results about P = 23 and 37 were wrong, and Eu proved P in 1738. The result of 29 = 29 is also wrong, and he proves that the conclusion about P = 31 is correct. It is worth pointing out that Qatardi has made his conclusions with a manual one, and Faima and Eu have used the most advanced mathematical knowledge in the time, avoiding many complex calculations and therefore possible mistake. Marin Mersenne, French, delivered his results in 1644. He claims that it is the number of p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, 2p-1, and for other prime numbers of less than 257, 2P-1 are all . Today, we call the number of m_p = 2p-1 prime numbers, M_P m_p is the first letter of Mason's last name. It is quite difficult to use it to judge whether a large number is quite difficult. Mason's priest also admits that his calculation is not necessarily accurate. After a century, in 1750, Orapa announced that the mistakes of Mason priests were found: M_41 and M_47 were also the number of prime. But great, such as Eura, will also make a calculation error - in fact, M_41 and M_47 are not the number of prime. However, this is not the result of Mason's priests is right. To wait until 1883, Mason's priest announced that after more than two hundred years, the first error was found: M_61 is a prime number. The other four errors are then found: M_67 and M_257 are not the number, while M_89 and M_107 are the number of prime. Until 1947, the correct results of Mason M_P of P <= 257 were determined, that is, when P = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, and 127, M_P is the number of prime. Now this table has been repeatedly verified, there will be no mistakes. Here is a list of all Mason's numbers we now know: (We note that Mason's priest is not above - this kind of number has been named by his name, and the honor is confirmed.
) The number of serial number P M_P corresponds to confirm that the number of people perfect person is 1 2 1 1 ---- ---- 2 3 1 2 ---- ---- 3 5 2 3 --- - - - 4 7 3 4 ---- ---- 5 13 4 8 1456 Anonymous 6 17 6 10 1588 Cataldi7 19 6 12 1588 Cataldi8 31 10 19 1772 Euler9 61 19 37 1883 Pervushin10 89 27 54 1911 Powers11 107 33 65 1914 Powers12 127 39 77 1876 Lucas13 521 157 314 1952 Robinson14 607 183 366 1952 Robinson15 1279 386 770 1952 Robinson16 2203 664 1327 1952 Robinson17 2281 687 1373 1952 Robinson18 3217 969 1937 1957 Riesel19 4253 1281 2561 1961 Hurwitz20 4423 1332 2663 1961 Hurwitz21 9689 2917 5834 1963 Gillies22 9941 2993 5985 1963 Gillies23 11213 3376 6751 1963 Gillies24 19937 6002 12003 1971 Tuckerman25 21701 6533 13066 1978 Noll & Nickel26 23209 6987 13973 1979 Noll27 44497 13395 26790 1979 Nelson & Slowinski28 86243 25962 51924 1982 Slowinski29 110503 33265 66530 1988 Colquitt & Welsh30 132049 39751 79502 1983 Slowinski31 216091 65050 130100 1985 Slowinski32 756839 227832 45563 1992 Slowinski & Gage33 859433 258716 517430 1994 Slowinski & gage34 1257787 37 8632 757263 1996 Slowinski & Gage35 1398269 420921 841842 1996 GIMPS36 2976221 895932 1791864 1997 GIMPS37 3021377 909526 1819050 1998 GIMPS ?? 6972593 2098960 4197919 1999 GIMPS Is there an infinite number of Mersenne primes it? Mathematics can't answer this question. Third, look for a bigger number of prime numbers to find Messen pixels? Why break the record of the most known maximum number? What is this? If you are using it, you have to create a material wealth directly, then I have to tell you that there is no use of Messen's number, knowing a very large number of numbers seems to be used. Even if we know a number of huge Mason, we will not increase our wallet (, etc.! If you are only interested in the money, please don't immediately leave my article. I am actually Say, I have to rule out the 100,000 dollar bonus I mentioned in this article - your wallet may be drums. So please patience). But humans do not only need material wealth. What is the use of diamonds in the museum? Why is human beings to collect them? Because they are beautiful and rare. As the crystallization of human wisdom, the number, Mason's number and the perfect number of closely related to it are very beautiful.
Their definition is simple, but so mysterious, like Euclid, Cartier, Felma, Leibniz, and Eura, is a large number of studies of it because of their beauty; Everyone also saw that after more than 2,000 years, we have only collected 38 Mason who have been collected by countless generations. They are very rare. For mathematicians, the number of collected prime, Mason's number and the perfect number are what is as fun as collecting diamonds. Humans also need glory - maybe more than wealth. In sports, you can run more faster, have a higher jump, is there really a practical material use? No, we like to accept challenges, we hope to win. Breaking a sports world record, climbing the Everest, driving the boat crossing the Pacific ..., it is a challenge to the limits of human physical strength; and find a larger number of prime, is a challenge to human wisdom. When we have completed an unprecedented task, we will always feel very proud. In 1963, when the 23 Mason's number was found, it was found that its Nursa University of Illinois is so proud, so that all the letters from the department are knocked "211213-1 is a single number" Postmark. After the Euca proves M_31, the next maximum record was obtained by Landry in 1867: M_59 / 179951 = 3203431780337. This is not a number of Mason. This record has been maintained for nine years. In 1876, Edward Lucas used a more advanced method than Fayman and Euler, which proved that M_127 is a number of prime. This record was maintained for seventy five years. Until Ferrier used a hand-shake computer in 1951 (2148 1) / 17 is a prime number, it has 41 digits. By means a hand-shake computer, it is probably the problem that can be discussed by the method of handling computer. However, the development of technology has become unnecessary. It is worth pointing out that in the course of mankind, the improvement in mathematics theory is much more important than the powerful and tough computing capacity. Lucas's approach was simplified in 1930, Lucas Leme test became a standard method for the number of Mason. (Lucas-Leme Test: For all odd numbers P, m_p larger than 1 is the number of prime numbers and only when M_P is eliminated S (P-1), where s (n) is from s (n 1) = s (n) 2-2, S (1) = 4 Recursive definition. This test is particularly suitable for computer operations, as the operation of M_P = 2P-1 can be implemented simply with the transposition and addition operation of the computer. Judging a number of Mason is the number of prime numbers than that of the other type of similar size. It is necessary to make much simple, so most records are Mason, which is Mason. In 1951 Miller and Miller & Wheeler uses the EDSAC computer (this computer is not as good as the general calculator we use now, it has only 5K memory) found that the number of 79-bit prime numbers 180 (m_127) 2 1. This record still has not been maintained. The Robinson applied SWAC computers, found 13 and 14 Mason: M_521 and M_607 in early 1952, and the number of three consecutive Mason is also found in the same year: M_1279, M_2203 and M_2281. In the later generation, the computer used in order to break the record of huge prime records is getting stronger and more powerful, including the famous IBM360 computer, and supercomputer CRAY series. You can refer to the Mason population table to understand this competition process. At this time, only one of the number of "known biggest prime numbers" is sitting on the number of prime numbers, which is 39158 * 2216193-1, found in 1989.
The M_1257787 discovered in 1996 was the last Mason popset found by the supercomputer, and the mathematician used the Cray T94. Then, GIMPS is coming. Fourth, GIMPS - Internet Mason Sui Search 1995 Program Designer George Waterman began to collect data about Mason's calculation. He has prepared a Mason Sui Looking Program and put it on the web for free use of math enthusiasm. This is the "Internet Mason Sui Search" program (GIMPS, The Great Internet Mersenne Prime Search). In this plan, more than a dozen mathematics experts and thousands of mathematics enthusiasts are looking for the next maximum of Mason, and check the gaps that have not been explored between the previous Mason records. For example, in the Mason table table, the last number of the number is unknown. We don't know if Mason's number of Mason 3, and there is still other unrecognized Mason numbers. 1997 Scott Kurowski and others have established a "PrimeNet" to send allocated search intervals and send reports to GIMPS. Now as long as you go to GIMPS homepage to download the free app, you can immediately participate in the GIMPS plan to search for Mason. Almost all common computer platforms have available versions. The program runs on your computer at the lowest priority, so there is almost no impact on your usual use of your computer. The program can also be stopped at any time, which will continue to calculate from the stop when starting. From 1996 to 1998, GIMPS plans to discover three Mason: M_1398269, M_2976221 and M_3021377, are the results obtained using a Pentium-type computer. In March 1999, an "EFF, ELECTRONIC FOUNDATION)" EFF, ELECTRONIC FOUND) is funded by anonymous personnel to find a bonus for huge prime numbers. It rules a $ 50,000 bonus to the first individual or agency that finds more than one million prime numbers, which is the bonus of our most open Haguratala. The latter bonus is: more than 10,000, 100,000 US dollars; more than 100 million, $ 190,000; more than 1 billion, $ 250,000. The verification of the results of the search results is very stringent. For example, the result must be explicit - you can't claim that your result is a solution of equations with one hundred equations, but do not solve it. The result must be verified independently of another computer. All of these rules explain on the EFF website. It should be noted that the hope of obtaining a bonus by participating in the GIMPS program is quite small. The computer used by Hajaratwara is one of 21,000 computers at the time. Every participant is verifying the number of different Mason numbers allocated, of course, most of them are not prime numbers - only about 30,000 points of possibilities encountered a number of prime numbers. The next 100,000 dollars will be issued to the first individual or institution that finds more than 10 million copies. This time calculation will be approximately 125 times the last one. Now GIMPS is calculated to be 70 billion floating point operations per second, and the most advanced super vector computers today, such as the operation capacity of CRAY T932. But if GIMPS wants to use such a supercomputer, you will need to pay about $ 200,000 a day. And now what they need, just the cost of supporting website operation, and a total of millions of dollars a bonus. 5. Online distributed computing plans GIMPS is only a introduction to these plans on the Internet on the Internet on the Internet, and the GIMPS homepage has introduced. Distributed computing is a computer discipline, which studies how to divide a very large computing ability to solve the problem into many small parts, then assign these parts to many computers, and finally integrate these calculations to get the final the result of. Sometimes the amount of computation is so large, you need to work together around the world to work together, in order to get the results in reason.
