Mersenne prime is composed ofMason numberAnd come.
The so-called Mason number is like 2p- 1, where the exponent p isprime number, often recorded as Mp。If Mason number is prime, it is called Mason prime.
useFactorizationIt can be proved that if 2n- 1 is a prime number, then the exponent n is also a prime number;Conversely, when n is a prime number, 2n- 1 (i.e. Mp）It is not necessarily a prime number.Most of the first few smaller Mason numbers are prime numbers, but the larger the Mason number, the more difficult it is to appear.
Only 51 Mersenne primes were found, the largest being Meighty-two million five hundred and eighty-nine thousand nine hundred and thirty-three(i.e. 2eighty-two million five hundred and eighty-nine thousand nine hundred and thirty-three- 1), there are 24862048 bits.
Whether there are infinitely many Mersenne primes is one of the famous unsolved problems.
Chinese name
Mersenne prime
Foreign name
Mersenne prime
Alias
twop- Prime number of type 1
Title
Numerous pearls, diamonds in number theory, mathematical treasures, prime number kings, etc
prime numberIt means that only 1 and itself can be used in integers greater than 1to be divisible byThe number of.There are infinite prime numbers, but by the end of 2018, only 51 prime numbers can be expressed as 2p- 1 (p is prime), which is Mason prime (such as 3, 7, 31, 127, etc.).It is based on the 17th century French mathematicianMarin Mersenne Name.
Mason prime is an important content in number theory research, which dates back toancient GreekPeople will know that 2pThe concept of prime number of type - 1.Because this prime number has unique properties (andPerfect numberFor thousands of years, it has attracted many mathematicians and countless mathematical enthusiasts to study and explore it.
In modern times, the in-depth exploration of Mason primes has promoted the development of various disciplines and new technologies.It is also human curiosityThirst for knowledgeandSense of honorThe best witness of.
origin
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As early as 300 BC,ancient GreekmathematicianEuclidAnd started research 2pFirst of all, he is in the famous book《Geometric primitives》Chapter 9 discussesPerfect numberAnd 2pThe relationship between prime numbers of type - 1.
In June 1640, Fermat was givingMarin Mersenne (Marin Mersenne) wrote in a letter: "In the hard research of number theory, I found three very important properties, which I believe will become the basis for solving the problem of prime numbers in the future."p- 1.
Marlin Mason was a unique central figure in European scientific circles in the 17th century. He often kept in communication with many scientists, including Fermat, to discuss mathematics, physics and other issues.In Europe at that time, academic journals andscientific research institutionIt is far from there, and there is no such form of international conference. Mason, who has extensive contacts, enthusiasm, sincerity and erudition, has become a bridge between scientists from all over the world. Many scientists are happy to tell him their research results, and then he will tell them to more people.Mason orAcademy of France The founder ofHistory of ScienceOne of the most important scientists in the world.[1]
Marlin Mason and 2p-1
Mason's research on 2pNumerals of type - 1 have been calculated and verified, and in 1644, in his book Random Thoughts on Physical Mathematics, he asserted that in prime numbers ≤ 257, when p=2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, 2p- 1 is a prime number, and others are composite numbers.The first seven numbers (that is, p=2, 3, 5, 7, 13, 17, 19) have been confirmed by predecessors, while the last four numbers (that is, p=31, 67, 127, 257) are Mason's own inference.Because of Mason's high academic status in the scientific community, people at that time believed his assertions.
Later, people realized that Mason's assertion actually contained a number of errors and omissions.However, Mason's work has greatly inspired people to study 2p- The enthusiasm of type 1 prime numbers has freed them from their subordinate status as "perfect numbers". It can be said that Mason's work is 2pA turning point and milestone in the study of type 1 prime numbers.Because Mason is knowledgeable, talented, enthusiastic and the earliest systematic researcher 2pThe number of type - 1, in memory of him, is called“Mason number”, and MpWrite it down (where M is the first letter of Mason's name), that is, Mp=2p－1。If Mason number is a prime number, it is called "Mason prime number" (that is, 2p- type 1 prime number).
Look for the journey
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For more than 2300 years, only 51 Mersenne primes have been found by human beings. Because of the rarity and charm of these primes, they are known as the "pearls of the sea".Since Mason put forward his assertion, people have found thatMaximum prime numberAlmost all are Mersenne primes, so the process of finding a new Mersenne prime is almost the same as the process of finding a new maximum prime.
The exploration of Mason prime is extremely difficult, which requires not only profound theory and skilled skills, but also hard calculation.
Era of manual calculation record
As early as the 4th century BC, Euclid (330-275 BC), a famous mathematician in ancient Greece, proposed thatpThe concept of prime number of type - 1 can be expressed as 2pA prime number in the form of - 1.He found that there was a close relationship between this type of prime number and perfect number: if 2p- 1 is a prime number, then 2p－1（2p- 1) is a perfect number.Euclid's conclusion is 2pThe study of type - 1 prime numbers has laid a foundation.However, in BC, when the computing power was low, people only knew four 2'sp- Prime number of type 1:
、
、
and
, the discoverer has no way to verify.
Euclid started the initial journey of exploration
The fifth 2 in the 15th centuryp- Prime number of type 1
The discoverer of.
For a long time, people thought that all 2pThe numbers of type - 1 may all be prime numbers, but Regius corrected this wrong view in 1536.He pointed out that Meleven=23 × 89 is not a prime number.So people began to think deeply about what 2pIs the number of type - 1 prime?How many prime numbers are there?Human Search 2p- The road of type 1 prime numbers is really on track.
Qatari Di became the earliest discoverer in history
First on 2p- Type 1 numbers are sortedItalyMathematician Pietro Cartaldi (1548-1626).In 1588, Qatari Di correctly pointed out that p=17 and 19,2p- 1 is a prime number;But he later proposed that p=23, 29, 31 and 37, 2p- 1 are also prime numbers.In the age of Qatari Di, judge 2pIt is extremely difficult to determine whether the number of type - 1 is prime.Although Qatari Di's conclusion has been verified by later generations to be three wrong, people still regard
and
These two prime numbers belong to his discovery.
Marlin Mason put forward the famous "assertion"
In the 17th century, French mathematician Marlin Mason (1588~1648) compared 2pThe number of type - 1 has been studied more comprehensively and deeply.In 1644, Mason put forward his four "2"p- Type 1 prime number: Mthirty-one、Msixty-seven、Mone hundred and twenty-sevenAnd Mtwo hundred and fifty-sevenThis is the famous "Mason assertion".Mason died four years after making the "assertion".Later, people found many mistakes and omissions from Mason's assertions, and did not put any 2pPrime of type - 1“Right of discovery”Belongs to him.However, in memory of MasonpThe pioneering work done in the study of type - 1 primes has since been called "Mason primes".
Fermat and Euler successively proved that M23, M29 and M37 are composite numbers
In the era of manual calculation, every step forward is particularly difficult.In 1772,SwitzerlandMacromathematicianLeonhard Oura(1707~1783) In the case of blindness, it was proved by mental calculation
It is indeed a prime number.This is the eighth Mason prime number found by people. It has a total of 10 digits, which is the largest prime number known in the world at that time.Euler also proved Euclid'sInverse theorem: All even perfect numbers have 2p－1（2p- 1), of which 2p- 1 is a prime number.This shows that Mason prime and even perfect numbers areOne-to-one correspondenceOf.
Euler also made outstanding contributions to Mason primes
The study of Mersenne primes has made new progress 100 years later.French mathematician in 1870sEdward Lucas (1842 ～ 1891) proposed a method to distinguish MpWhether it is an important theorem of prime numbers Lucas theorem provides a powerful tool for Mason to find prime numbers.[2]In 1876, Lucas proved that
It is indeed a prime number, which is the largest Mason prime number found by manual calculation, up to 39 digits long.
Lucas Promotes the Study of Mason Prime
So far, two of Mason's assertions have been correct.However——
From the end of the 19th century to the beginning of the 20th century, people successively found three Mason primes by using Lucas theorem.In 1883,RussiaMathematician Ivan Bofosin (1827 ～ 1900) proved that
It is also a prime number - this is what Mason missed.Mason also missed the other two prime numbers
and
They were discovered by American amateur mathematician Ralph Bowers (1875-1952) in 1911 and 1914 respectively.
Cole found the factor of M67
Mason's assertion has two mistakes.In 1876, Lucas was the first person to deny "Msixty-sevenIt is a prime number, which has been believed by people since Mason's assertion, but he has not found its factor.It was not until 1903 that the mathematician Kohl (1861-1926) calculated Msixty-seven=193707721×761838257287。In 1922, mathematician Kletchek (1882-1957) verified that Mtwo hundred and fifty-sevenIt is not a prime number, but a composite number.
In the long years of manual calculation, people went through hardships and only found 12 Mersenne primes.
Computer Age
In the 1930s, American mathematician Derek Lemmer (1905-1991) improved Lucas' work and gave apThe new primality test method ofLucas Lemmer test：Mp>3 is a prime numberif and only ifLp－2=0, where Lzero=4，Ln+1= ( Lntwo－2 ) modMp。This method is very suitable for computer computing, so it has played an important role in the "computer age".
Lemmer improved Lucas' method
On the evening of January 30, 1952, American mathematician Raphael Robinson (1911~1995) compiled this method into a computer under the guidance of Lemmerprogram, usingUCLASWAC computer, found two Mason primes with more than 100 digits in a few hours
and
。SWAC, the standard western automatic computer, was one of the fastest computers at that time.In the following months, Robinson used the computer to find
、
and
。
Robinson began to use the computer to find Mason primes
The emergence of computers has greatly enhanced the search for Mason primes, and scientists and amateur researchers from all over the world have joined in the search for Mason primes.They competed with each other and enjoyed it, which gave birth to many new discoveries.In September 1957, the 18th Mersenne prime
coverSwedenMathematician Hans Lissel (1929~2014) used the BESK computer to find;In November 1961,
and
Used by American mathematician Alexander Herwitz (1937 ～)IBM-The 7090 computer was found on the same day, and the two Mason primes were still discovered by the University of California, Los Angeles.
Gillis used a mainframe computer to find three Mason primes
In May 1963, American mathematician Donald Gilles (1928~1975) usedMainframeContinuous Find
and
。On the evening of June 2, 1963, when Gillis found the 23rd Mason prime
When,abcThe normal program broadcast was interrupted, and this important news was released at the first time.The United States that discovered this prime numberUniversity of Illinois All the teachers and students of the Department of Mathematics are very proud. In order to let people all over the world share this achievement, they have typed "M" on all the letters sent from the Departmenteleven thousand two hundred and thirteenIt's a primepostmark。
M11213 postmark
In the mid-1960s, with IBM-360 as the representativeThird generation computerThe emergence of "" speeds up the search for Mason primes, but with the increase of the exponent p value, the generation of each Mason prime becomes more difficult.American mathematician on the evening of March 4, 1971Bryant Tuckman(1915 ～ 2002) Using IBM-360/91 computer, we finally found a new Mason prime after nearly 8 years
。U.S.ACBSThis discovery was also reported at the first time.
By the end of October 1978, almost all major news organizations in the world (including China's Xinhua News Agency) had rushed to report the following news: two 18 year oldsAmerican High SchoolLangdon Knoll and Nichol ran their own calculation program on the Cyber-174 computer and found the 25th Mason prime after 350 hours of continuous calculation
。
Noel's discovery arouses worldwide attention
In February of the next year, Noel made persistent efforts to find the 26th Mason Prime
。
With the development of mathematical theory, the computer functions used to find Mason prime numbers are becoming more and more powerful, including the famousSupercomputerCraySeries.On April 8, 1979, David Slowinsky and Nelson, computer experts of Cray Corporation, found more than 10000 Mason primes using Cray-1 computer
；Schlossky himself used an improved Cray XMP computer and found it again from 1982 to 1985
、
and
, but he was not sure whether there was any difference between Mone hundred and thirty-two thousand and forty-nineMersenne prime of.
"Prime Number King" Slovinsky
In January 1988, Kerquette and Welsh used NEC-SX2 super high speedparallel computer Sure enough, I found the "fish that missed the net"
。After four years of silence, a research group of Havel Laboratory (the British Atomic Energy Technology Authority) suddenly announced that they had found a new maximum prime number on February 19, 1992
, the number of digits exceeds 200000.In order to achieve this result, they ran the computer for more than two years and spent 120000 pounds.It seems that Mason's quest for primes is getting farther and farther away from ordinary people······
Previously knownMaximum prime number391581 × 2 discovered in 1989two hundred and sixteen thousand one hundred and ninety-three- 1, not Mason prime.Mseven hundred and fifty-six thousand eight hundred and thirty-nineThe discovery of Mersenne primes announced that Mersenne primes were once again on the throne of the largest known primes.Since then, this throne has been firmly occupied by Mason Prime, and has never been sidelined.
Zhou Haizhong proposed "Zhou's Guess"
While people are looking for Mersenne primes, they are also studying the distribution law, which is an important property of Mersenne primes.Few Mersenne primes have been found, and people do not know their infinity, so it seems more difficult to explore its distribution than to find new Mersenne primes.Mathematicians have put forward some suggestions onMersenne prime distributionBut their guesses are similarexpressionIt is given, and the closeness to the actual situation is not satisfactory.Chinese scholarsZhou HaizhongAccording to the known Mersenne primes and their permutations, a conjecture about the distribution of Mersenne primes was also put forward in February 1992, and the exact expression of its distribution was given for the first time, which attracted people's attention.[3]This conjecture was later named“Zhou's conjecture”。
Found Cray-T94 computer of M1257787
On January 4, 1994, Schlossky and Gage regained the crown of discovering the largest known prime number for Cray again - this prime number is
。And the next Mersenne prime
It is still their achievement.Slovinsky is known as the "king of prime numbers" because of his discovery of seven Mason primes.M discovered in September 1996one million two hundred and fifty-seven thousand seven hundred and eighty-sevenIt is the last Mersenne prime discovered by supercomputers so far. The researchers used Cray-T94, which is also the 34th Mersenne prime discovered by humans.
Internet era
It is too expensive to use supercomputers to find Mason primes, and the number of people who can participate is limited. The emergence of grid, a new technology, makes the search for Mason primes return to the mass era of "everyone participates".In the middle and late 1990s, with the joint efforts of American programmers Watman and Kurowski, the world's first Internet basedDistributed ComputingProject - Internet Mason Prime Search（GIMPS）。As long as people download a free program to calculate Mason primes on the homepage of GIMPS, they can immediately participate in the project to search for new Mason primes.
George Waterman, founder of GIMPS project
From 1996 to 1998, the GIMPS project found three Mason primes:
、
and
The discoverers came from France, Britain and the United States.
June 1, 1999, United StatesMichiganMathematician Najan Hajaratwara of Plymouth found the 38th Mason Prime through the GIMPS project
。This is the last Mason prime number discovered in the 20th century, and the first prime number known to exceed 1 million digits.Hajjaratwara uses only an ordinary Pentium II computer.Although the performance of this computer is not high, distributed computingnetwork connectionsAt that time, there were tens of thousands of ordinary computers in the world, so the total computing speed was still equivalent to that of supercomputers.
The last discoverer in the 20th century, Hajilatwala
Cooper and Boone Cooperate to Discover the 43rd and 44th Mersenne Prime
Entering the 21st century, withpersonal computerWith the further popularization of and the improvement of computing speed, people have found many larger Mersenne primes.On November 14, 2001, a 20-year-old Canadian youth Michael Cameron found
, opening the prelude to the search for Mason primes in the new century.[4]Since then, from 2003 to 2006, the GIMPS project has found five Mason primes:
、
、
、
and
The record of the known maximum prime number is getting closer to the 10 million bit mark.[5-9]
Smith Adds New Discovery to UCLA
August 23, 2008, United StatesUCLAEdson Smith, a computer expert at, finally found more than 10 million Mason primes
。[10]This is the eighth Mason prime discovered by the school.It has 12978189 digits. If you use ordinary font size to print this super large prime number continuously, its length can exceed 50 kilometers!This achievement was recognized by the United States《times》The magazine was rated as one of the "50 best inventions" of the year, ranking 29th.[11]The discoverer Smith also won a $100000 award from the EFF.
In the following year, two smaller Mason primes were found by German and Norwegian volunteers.[12-13]
Only two weeks after Smith's discovery
Compared with the prime number discovered by Smith, the difference is only 140000 digits.This is another discovery of Mason primes "out of sequence" after 1988.
Cooper has found "the largest prime number" many times
In 2013 and 2016, the United StatesUniversity of Central Missouri professor of mathematicsCurtis Cooper Using more than 800 computers in the campus, two Mason primes were found in succession
and
。[14-15]The latter was actually calculated by computer as early as September 2015, but people didn't notice it until January 2016, which delayed the discovery date of this Mason prime by more than three months.Cooper discovered four Mason primes in total through the GIMPS project, becoming the new "king of prime numbers".Cooper's success has set off a new upsurge of exploring Mason primes around the world.
Pace was lucky to find the 50th Mason prime
On December 26, 2017, Jonathan Pace, an American volunteer who has been involved in the GIMPS project for 14 years, found the 50th Mason prime number known to mankind
。[16]To commemorate this landmark discovery, a Japanese publishing house also released a book called《The largest prime number in 2017》Books.[17]However, only one year later, the record of the largest known prime number was refreshed again.The new record is
，By the United StatesFloridaOkaraPatrick Laroche found it on December 7, 2018.[18]This is the 17th Mersenne prime discovered in the 23 years of the GIMPS project.
The 50th Mason prime is printed into a book
On October 6, 2021, Mfifty-seven million eight hundred and eighty-five thousand one hundred and sixty-oneAfter more than 8 years, this number has been determined as the 48th Mersenne prime.[19]
Mason prime table
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By December 2018, a total of 51 Mersenne primes had been found.Now, the values, digitsDiscovery time、DiscovererThe list is as follows:
Each table lists the Mason prime numbers discovered manually, by computer and through the GIMPS project.
It is uncertain whether there is still an unknown Mersenne prime between M48 and M51, and the subsequent serial numbers are marked with *.
The values of Mersenne prime numbers in the last two tables are omitted.
GIMPS project
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In early 1996, American mathematicians andProgrammerGeorge Woltman (George Woltman) prepared aPrime95And put it on the website for mathematicians and math enthusiasts to use for free“Internet Mersenne Prime Search”Project, short forGIMPS。The project adoptsGrid computingMode, using a large number of ordinary computersIdle timeTo get the equivalent ofSupercomputerOfComputing power。In 1997, American mathematician and programmer Scott Kurowski and others established the "PrimeNet" to automate the allocation of search intervals and the sending of reports to GIMPS.A huge database records the allocation of all tasks and calculation reports. If a returned calculation report shows that a new Mason prime number has been found, it needs to be verified by a special agency with different algorithms before it can be officially confirmed.
EFF awards prizes to winners (first from the right)
In order to motivate people to find Mason primes and promoteGrid technologyDevelopment, headquartered in the United StatesSan FranciscoOfElectronic Frontier Foundation(EFF) announced to the world in March 1999 that it was established to search for new and larger Mason prime numbers through the GIMPS project“Collaborative computingAward ".It stipulates that 50000 US dollars (already issued) should be awarded to the first person or institution that finds more than 1 million digits.The following bonuses are in turn: more than 10 million, 100000 US dollars (already awarded);More than 100 million, 150000 dollars;More than 1 billion, 250000 dollars.[20]In addition, the discoverer of Mason primes can also get at least $3000 from GIMPS.But in fact, most volunteers participate in the project not for money, but for fun, honor and spirit of exploration.
At present, people have found 17 Mason primes through the GIMPS project, whose discoverers are from the United States, Britain, France, GermanyCanadaandNorway。There are more than 190 in the worldcountries and regions More than 600000 people participated in thisInternational cooperation projectsAnd used millions of computers（CPU）Networking to find new Mersenne primes.[21]Of this projectComputing powerIt has exceeded the computing power of any of the most advanced super vector computers in the world,Operation speedIt can reach 850 trillion times per second.Famous《natural》The magazine said: The GIMPS project will not only further stimulate people's enthusiasm for finding Mason prime numbers, but also arouse people's interest inGrid technologyGreat importance is attached to applied research.
significance
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Mason prime has been an important part of number theory since ancient times. Many mathematicians in history have studied this special form of prime numbers.From the ancient Greek era until the 17th century, it seems that the purpose of people looking for the meaning of Mason primes is to find perfect numbers.But since Mason put forward his famous assertion, especially since Euler proved the inverse theorem of Euclid's complete number theorem, even perfect numbers have only been a "by-product" of Mason's prime numbers.
The search for Mersenne primes has a very rich meaning in contemporary times.Finding Mersenne prime is the most effective way to find the largest known prime number.Self Euler proof Mthirty-oneSince it was the largest prime number at that time, Mason's prime number almost included all the champions in the world competition in which the largest prime number was found.
Finding Mason prime is a test computerOperation speedAnd other functions, such as Mone million two hundred and fifty-seven thousand seven hundred and eighty-sevenIt was obtained in September 1996 when Cray tested the computing speed of its latest supercomputer.Mason prime has played a unique role in promoting the improvement of computer functions.It is found that Mason prime numbers need not only high function computers, but also prime number discrimination andnumerical calculationIts research has promoted the development of number theory, the "queen of mathematics"Computational mathematicsAnd the development of programming technology.
Distributed computing is the main way to find Mason prime at present
The latest meaning of searching for Mason prime is that it promotesDistributed ComputingTechnology development.From the fact that the latest 17 Mason primes were discovered in the Internet project, we can imagine the power of the network.Distributed Computing TechnologyIt is possible to use a large number of personal computers to do projects that can be completed only by supercomputers. This is a very promising field.Its exploration also promotedfast Fourier transform Application of.
Mason primes are also useful in practical fields, and people have used large primes in modern cryptographyDesign field。The principle is that it is very difficult to decompose a large number into the product of several prime numbers, but it is much easier to multiply several prime numbers.In this kind of password design, a large prime number is required. The larger the prime number, the less likely the password will be cracked.
Because the exploration of Mason prime numbers needs the support of various disciplines and technologies, and because of the international repercussions caused by the discovery of new "big prime numbers", the research ability of Mason prime numbers has, in a sense, marked the scientific and technological level of a country, rather than just representing the research level of mathematics.Top British scientistsOxfordProfessor Marcos Sotoey once said that the research progress of Mason primes is an important aspect of scientific development, and people expect a modern Euclid to prove that Mason primes will never die.
Mason prime, a shining pearl in the ocean of mathematics, is attracting more aspiring people to look for and study it with its unique charm.
Theoretical exploration
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In December 2017, Zhang Miaobao discovered the distribution law of Mason primes and gave a more accurateCalculation formula。
The Calculation Formula of Mason Prime
3*5/3.8*7/5.8*11/9.8*13/11.8*......*P/(P-1.2)-1=M
P is the exponent of Mason number, and M is the number of Mason primes below P.
The following are the calculated values andActual numberSituation of:
Index 5, calculation 2.947, actual 3, error 0.053;
Index 7, calculation 3.764, actual 4, error 0.236;
Index 13, calculation 4.891, actual 5, error 0.109;
Index 17, calculation 5.339, actual 6, error 0.661;
Index 19, calculation 5.766, actual 7, error 1.234;
Index 31, calculation 6.746, actual 8, error 1.254;
Index 61, calculation 8.445, actual 9, error 0.555;
Index 89, calculation 9.201, actual 10, error 0.799;
Index 107, calculated 9.697, actual 11, error 1.303;
Index 127, calculated 10.036, actual 12, error 1.964;
Index 521, calculated 13.818, actual 13, error -0.818;
Index 607, calculation 14.259, actual 14, error -0.259;
Index 1279, calculated 16.306, actual 15, error -1.306;
Index 2203, calculation 17.573, actual 16, error -1.573;
Index 2281, calculation 17.941, actual 17, error -0.941;
This formula is based on the distribution law of Mason primes.One is the first in ten thousand, and one is excluded, so subtract one.Without considering the overlap problem, P minus 1 should be enough. The overlap problem has been considered here, so the index of P minus 1.2 in Mason number gradually increases. Whether 1.2 is appropriate still needs to be tested in practice.
be-allOdd prime numberAre both factor numbers and Mason prime numbers of quasi Mason number (2 ^ N-1), thenMason composite numberThe number of factors of is only a part of the prime number.
In the sequence of 2 ^ N-1, a prime number first appears in the number of exponent N as a prime factor, and this prime number repeatedly appears in the sequence of 2 ^ N-1 as a factor number with N as the cycle.In this series, the index iseven numbersIs equal to three times four times the number of pyramids.
In the 2 ^ N-1 number sequence, if the index is greater than 6, except for Mason prime numbers, there is one or more new prime numbers as factor numbers. The factor number of the new Mason composite number minus 1 can be determined by this indexto be divisible by。
The factor number of a Mason composite appears in a Mason composite for the first time: and in the quasi Mason composite after the index of the Mason composite appears repeatedly,
One is the prime number of Mason prime number, which is never the factor number of Mason composite number.
One is the prime number of the factor number of the preceding Mason composite, which will never be the factor number of the following Mason composite.
The factor number of all Mason composite numbers minus 1 can be divided by the index of this Mason composite number, and the quotient is even.
A prime number appears as a factor number for the first time in a quasi Mason number that is not a Mason composite number. The prime number minus 1 can be divided by the index of the quasi Mason number, but there are exceptions. Quotients are not necessarily even numbers.
Mason prime numbers are all in [4 ^ (1-1)+4 ^ (2-1)+4 ^ (3-1)+...+4 ^ (n-1)] * 6+1 sequence, and the number of inclusion symbols is temporarily called quadruple pyramid number.
A prime number that is a factor number of four times the number of pyramids is not a factor number of Mason composite numbers.
At 4 ^ (1-1)+4 ^ (2-1)+4 ^ (3-1)++The numbers in the 4 ^ (n-1) sequence are all Mersenne primes if they are not equal to 6NM+- (N+M) multiplied by 6 and 1.
At 2 ^ P-1square rootThe following prime numbers appear as prime factors in the previous quasi Mason numbers. If there is no factor number for this Mason number, it must be a Mason prime number.But itsInverse theoremIt is not tenable.If the prime number has not appeared in the previous quasi Mason number, it may also be the factor number of Mason composite number.
The factors of Mason composite numbers are prime numbers of 8N+1 and 8N-1 forms.
On Mersenne Prime
In the 2 ^ n-1 sequence where the exponent n is infinite, Mason numbers and Mason primes only account for a small proportion.
according toFermat's small theoremEvery odd prime number will appear in the 2 ^ n-1 sequence as a number factor, but some appear earlier and some appear last.Only Mersenne prime first appeared in this sequence.Other prime numbers will not appear first, and the prime number that appears last is in the number minus 1, which is the place of Fermat's small theorem.
Every odd prime number regularly appears in the 2 ^ n-1 sequence as a factor number. A prime number appears in the 2 ^ n-1 sequence for the first time (including Mason prime number), and this prime number repeatedly appears in the 2 ^ n-1 sequence with n as the cycle. For example, 3 appears in n=2 for the first time, and there are 3 factors whose exponent can be divided by 2;7 appears for the first time in n=3, and the exponential divisible by 3 has a factor number of 7;5 first appears in n=4, and the exponent can be divided by 4 and has a factor number of 5.A prime number appears in the 2 ^ n-1 sequence n, no matter whether n is a prime number or not, as long as all odd prime numbers less than n are used to sieve, the exponent n is in it.If the composite number overlaps the previous prime number, there is no need to re screen.
To screen all number factors in the 2 ^ n-1 sequence, you must use all prime numbers less than or equal to the square root of 2 ^ n-1 to screen, so that the remaining ones that are not screened are Mason prime numbers.
The number sequence of 2 ^ n-1 is infiniteNatural numberNo matter how many times you sift a fraction, there will always be unlimited.So there are infinitely many Mersenne primes.
according toFermat's small theoremEvery odd prime number will appear in the 2 ^ n-1 sequence as a prime factor, but some appear earlier and some appear later.
Every odd prime number appears for the first time in the number of the exponent n of the 2 ^ n-1 sequence, and this n is the prime numberCorresponding number, it appears repeatedly in the cycle of n.
It is known that all Mersenne primes appear in Mersenne numbers.As long as the Mersenne composite number in the Mersenne number is screened out, the remaining Mersenne prime numbers are left.
Expand the Mason sequence, starting from the corresponding number 2 of 3, 2 points one point;The corresponding number of 5 is 4, and 4 is a composite number, so no operation is required;The corresponding number of 7 is 3, at 3:1;The corresponding number of 11 is 10, which is a composite number without operation;The corresponding number of 13 is 12, and 12 is a composite number, so no operation is required;This way, the number before the index of Mason number can be screened.All Mason numbers that have been counted twice or more are crossed out, and the remaining Mason numbers that have only been counted once are all Mason prime numbers.
This method is difficult to find the corresponding number when the prime number is large.
