Chinese name
Mason number
Foreign name
Mason number
inaugurator
Euclid, Fermat, marin mersenne
The earliest creative time
More than 300 years BC
catalogue
1? Basic information
2? Introduction to history
Basic information
Mason number, also known as Mason number, refers to a positive integer with the shape of 2 p- 1, in which the exponent p is a prime number and is often recorded as Mp. If it is a prime number, it is called mersenne prime. ? [ 1]?
Mersenne prime is an important content in the study of number theory, and people have been exploring mersenne prime since ancient Greece. Because of its unique properties (for example, it is closely related to perfect numbers) and infinite charm, this prime number has always attracted many mathematicians (including Euclid, Fermat, Euler and so on). ) and countless math lovers have explored it for thousands of years. ? [2]?
A prime number, also called a prime number, is a number that can only be divisible by itself and 1, such as 2, 3, 5, 7, 1 1 and so on. 2300 years ago, Euclid, an ancient Greek mathematician, proved that there are infinitely many prime numbers, and proposed that a few prime numbers can be written in the form of "2 p- 1", in which the exponent p is also a prime number. Because this prime number has many unique properties and infinite charm, it has been attracting many mathematicians and countless amateur mathematicians to explore it for thousands of years.
/kloc-the famous French mathematician Mei Sen made a systematic and in-depth inquiry into the type of "2 P- 1" and made a famous conclusion (now called "Mason conjecture"). Because he was the central figure of European scientific community at that time and the founder of French Academy of Sciences, the prime number "2 P- 1" was called "mersenne prime" in the field of mathematics, and the rest numbers were called Mei Sen composite numbers.
Introduction to history
Editor? broadcast
Marin mersenne (1588–1648) was a famous French mathematician and friar in17th century, and was a unique central figure in the European scientific community at that time.
1640 in June, Fermat wrote in a letter to Mei Sen: "In the difficult study of number theory, I found three very important properties. I believe that they will be the basis for solving the prime number problem in the future. " This letter discusses numbers in the form of 2 p- 1 (where p is a prime number). As early as more than 300 years BC, Euclid, an ancient Greek mathematician, initiated the study of 2 p- 1. When discussing the perfect number in the ninth chapter of his masterpiece Elements of Geometry, he pointed out that if 2 p- 1 is a prime number, then (2 p- 1) 2 (p-65438).
On the basis of Euclid and Fermat's related research, Mei Sen has done a lot of calculation and verification work on 2 p- 1, and asserted in his book Random Thoughts on Physical Mathematics that for p=2, 3, 5, 7, 13, 17, for all other numbers less than 257, 2. The first seven numbers (namely, 2, 3, 5, 7, 13, 17, 19) belong to the confirmation part, which he obtained by sorting out the previous work. The last four numbers (3 1, 67, 127 and 257) belong to the guessing part. However, people are still convinced of its assertion.
Although there are some mistakes in Mei Sen's assertion, his work has greatly aroused people's enthusiasm for studying 2 P- 1 prime numbers and freed people from the vassal status of "perfect numbers". Mei Sen's work is a turning point and milestone in the study of prime numbers. Because Mei Sen is knowledgeable, talented and enthusiastic, he first studied the number 2 P- 1 systematically and deeply. In order to commemorate him, the mathematics community called this number "Mason number"; And write it down in Mp (where m is the initials of Mei Sen), that is, MP = 2 p- 1. If the Mason number is a prime number, it is called "mersenne prime" (i.e. a prime number of type 2 P- 1).
It is worth mentioning that French mathematician Lucas and American mathematician Remo made important contributions to mersenne prime's basic research. Lucas-Lemmer test named after them is the most famous method to detect mersenne prime. ? [3] In addition, Zhou Haizhong, a mathematician and linguist in China, gave an accurate expression of Mason's prime number distribution, which provided convenience for people to find mersenne prime. This research achievement is named "Zhou conjecture" internationally.
The research team led by mathematician Cooper of Missouri Central University in the United States discovered the 49th mersenne prime -274, 207, 28 1- 1 on October 7th, 2065438. This prime number is also the largest known prime number, with 223386 18 digits. This is the fourth time that Professor Cooper has discovered a new mersenne prime through the GIMPS project, which has set a new record for him. The last time he found the 48th mersenne prime 257,885,161-kloc-0/was at 20 13 1, with 17425 170.
Mersenne prime is of great significance and practical value in contemporary times. It is the most effective method to find the largest known prime number, and its exploration has promoted the research of the "queen of mathematics" number theory and promoted the development of computing technology, cryptography technology, programming technology and computer detection technology. ? [4] No wonder many scientists believe that mersenne prime's research results reflect a country's scientific and technological level to some extent. Max Seautois, chairman of the British Mathematical Association, even thinks that its research progress is not only a sign of human intellectual development in mathematics, but also one of the milestones of the whole scientific and technological development.
All odd prime numbers are factorials of quasi-Mason numbers (2 n- 1). Any prime number that is a factor of four times the pyramid number A will not be a factor of Mei Sen composite number in the future, so some prime numbers may be a factor of Mei Sen composite number.
Mersenne prime's calculation formula
3*5/3.8*7/5.8* 1 1/9.8* 13/ 1 1.8*......*P/(P- 1.2)- 1=M
P is the exponent of Mason number, and M is the number of mersenne prime below P. ..
The following are calculated values and actual numbers:
Index 5, calculation 2.947, actual 3, error 0.053;
Index 7, calculated 3.764, actual 4, error 0.236;
Index 13, calculated as 4.89 1, actual 5, error 0.109;
Exponent 17, calculated 5.339, actual 6, error 0.661;
Exponent 19, calculated 5.766, actual 7, error1.234;
Index 3 1, calculated 6.746, actual 8, error1.254;
Index 6 1, calculated as 8.445, actual 9, error 0.555;
Index 89, calculated as 9.20 1, is actually 10, with an error of 0.799;
Index 107, calculated as 9.697, actual 1 1, error1.303;
Index 127, calculation 10.036, actual 12, error1.964;
Index 52 1, calculated 13.8 18, actual 13, error-0.818;
Index 607, calculated as 14.259, is actually 14, with an error of-0.259;
Index 1279, calculation 16.306, actual 15, error-1.306;
Index 2203, calculation 17.573, actual 16, error-1.573;
Index 228 1, calculation 17.94 1, actual 17, error-0.941;
This formula is based on the distribution law of mersenne prime. The thousand digits are 1. If 1 is excluded, 1 will be subtracted. Regardless of the overlapping problem, it is enough to subtract 1 from P. Since the overlapping problem has been considered here, it is enough to subtract 1.2 from P. The exponent of Mason number is gradually increasing, and whether 1.2 is suitable or not has to be tested in practice.
All odd prime numbers are divisors of quasi-Mason numbers (2 n- 1), so the divisors of Mei Sen composite numbers are only a part of prime numbers.
In the sequence of 2 n- 1, a prime number as a prime factor first appears in the number of exponent n, and this prime number as a factor appears in the sequence of 2 n- 1 with a period of n.
The factor number of Ameson composite number only appears once in Ameson composite number.
One is mersenne prime, which has never been a factor of Mei Sen's composite number.
One is the factor of the former Mei Sen composite number, and it will never be the factor of the latter Mei Sen composite number.
The number factor of all Mei Sen complex numbers minus 1 can be divisible by the exponent of this Mei Sen complex number, and the quotient is even.
The order of mersenne prime is (1+4+16+64+) ...+4a) * 6+1(the number mentioned in the previous paragraph a). This number is temporarily called the quadruple pyramid number, and the code is a.
The number +4A in the sequence 1+4+ 16+64+ is a positive inequality (not equal to 6NM+-(N+M)) multiplied by 6, and 1 is the mersenne prime.
In the 2 n- 1 series, all the indexes with even numbers are 3A.
Prove that mersenne prime is infinite
In mersenne prime's 2 n- 1 series, the exponent n is infinite, and Mason number and mersenne prime only account for a small proportion.
According to Fermat's last theorem, every odd prime number will appear as a number factor in the 2 n- 1 sequence, but some will appear in advance and some will appear last. Only mersenne prime appeared in this series for the first time. Other prime numbers will not appear first, and the latest prime number is in this number minus 1, which is where Fermat's last theorem lies.
In the 2 N- 1 sequence, every odd prime number regularly appears in the form of a factor. When a prime number first appears in the 2 N- 1 sequence (including mersenne prime), this prime number appears repeatedly in the 2 N- 1 sequence with a period of n, for example, when 3 first appears in n=2, the exponent can be an integer multiple of 2. For example, when 7 first appears at n=3, there are all factors of 7, and its index can be divisible by 3; If 5 first appears in n=4, the exponent can be divisible by 4, and all the factors of 5 exist.
The above theorem can prove that mersenne prime is infinite without proof, because Fermat's last theorem is a guarantee.
A prime number appears in the sequence of 2 N- 1 n, no matter whether n is a prime number or not, as long as all odd prime numbers less than n are used for screening, the exponent n is among them. If the composite number overlaps with the previous prime number, there is no need to re-screen.
To filter out all the number factors in the sequence of 2 n- 1, it is necessary to use all the prime numbers less than or equal to the square root of 2 n- 1 to filter, thus leaving mersenne prime without screening.
The sequence of 2 n- 1 is infinite, and the infinite number of natural numbers is a fraction of how many times you filter it, and it will always be infinite. So there are infinitely many mersenne prime.
In 20 18, Jonathan Pace successfully found the 50th mersenne prime M772329 17 with 23,249,425 digits by "Internet mersenne prime search".