Suppose there are only a limited number of n prime numbers, which are arranged as p 1, p2, ..., pn, and let n = P 1× P2×...× PN, then is N+ 1 a prime number?
If N+ 1 is a prime number, then N+ 1 is greater than p 1, p2, ..., pn, so it is not in those assumed prime number sets.
If N+ 1 is a composite number, because any composite number can be decomposed into the product of several prime numbers; The greatest common divisor of n and N+ 1 is 1, so N+ 1 cannot be divisible by p 1, p2, ..., pn, so the prime factor obtained by this complex number decomposition is definitely not in the assumed prime number set.
Therefore, whether the number is a prime number or a composite number, it means that there are other prime numbers besides the assumed finite number of prime numbers.
For any finite set of prime numbers, the conclusion that a prime number is not in the assumed set of prime numbers can always be obtained by the above method.
So the original assumption doesn't hold water. In other words, there are infinitely many prime numbers.
Other mathematicians have also given their own proofs. Euler proved by Riemann zeta function that the sum of reciprocal of all prime numbers is divergent, Ernst Cuomo proved more concisely, and hillel furstenberg proved by topology. Fermat, known as "/kloc-the greatest French mathematician in the 7th century", also studied the properties of prime numbers. He found that if Fn = 2 (2 n)+ 1, then when n is equal to 0, 1, 2, 3 and 4 respectively, Fn gives 3, 5, 17, 257 and 65537 respectively, which are all prime numbers. Because F5 is too big (F5 is fermat number. However, F5 has a problem! 67 years after Fermat's death, the 25-year-old Swiss mathematician Euler proved:
F5 = 4294967297 = 641× 6700417, which is not a prime number, but a composite number!
More interestingly, mathematicians have never found out which Fn values are prime numbers, and they are all composite numbers. At present, due to the large square, there are few proofs. Now mathematicians get the maximum value of Fn: n= 1495. This is a super astronomical figure, with as many as 10 10584 digits. Of course, although it is big, it is not a prime number. In the17th century, there was a French mathematician named Mei Sen. He once made a guess: 2 p- 1, when p is a prime number, 2 p- 1 is a prime number. He checked that when p=2, 3, 5, 7, 17, 19, the values of the algebraic expressions obtained are all prime numbers. Later Euler proved that when p=3 1, 2 p- 1 is a prime number. When p = 2,3,5,7,2 p-1are all prime numbers, but when p= 1 1, the obtained 2047=23×89 is not a prime number.
Now there are three Mason numbers left, p=67,127,257, which have not been verified for a long time because they are too big. 250 years after Mei Sen's death, American mathematician Kohler proved that 267-1=193707721× 761838257287 is a composite number. This is the ninth Mei Sen number. In the 20th century, people successively proved that 10 Mason number is a prime number and 1 1 Mason number is a composite number. The disordered arrangement of prime numbers also makes it difficult for people to find the law of prime numbers.
The research team led by CurtisCooper, a professor of mathematics at the University of Central Missouri in the United States, found the largest known mersenne prime-2 57885 16 1 (that is, 257885161minus 65438), and this prime number has/kloc-0. If you print continuously with ordinary font size, its length can exceed 65km!
While people are looking for mersenne prime, the research on its important property-distribution law has been going on. Mathematicians in Britain, France, Germany, the United States and other countries have given guesses about the distribution of Mason prime numbers, but they are all given by approximate expressions, which are not close to the actual situation. Zhou Haizhong, a mathematician and linguist in China, is a leading figure in this field. He first gave the exact expression of Mason prime number distribution in 1992. This achievement was later named "Zhou's conjecture" internationally. Goldbach's Conjecture
Goldbach conjecture can be roughly divided into two kinds (the former is called "strong" or "double Goldbach conjecture" and the latter is called "weak" or "triple Goldbach conjecture"): 1, and every even number not less than 6 can be expressed as the sum of two odd prime numbers; 2. Every odd number not less than 9 can be expressed as the sum of three odd prime numbers.
Riemann hypothesis
Riemann conjecture is a difficult problem that has puzzled the mathematics field for many years. It was first put forward by German mathematician Bernhard Riemann, but so far no one has given a completely convincing and reasonable proof. That is, how to prove that "all solutions of the equation about prime numbers are on a straight line".
The prime number shaping in this prime number law, "all solutions of the equation about prime numbers are on a straight line", is transformed into a spherical prime number distribution.
twin prime conjecture
1849, Polinak put forward the conjecture of twin prime numbers, that is, he guessed that there were infinite pairs of twin prime numbers.
The "twin prime numbers" in the conjecture refers to a pair of prime numbers, and the difference between them is 2. For example, 3 and 5, 5 and 7, 1 1 3, 100 16957 and 100 16959 are all twin prime numbers.
100 16957 and 100 16959 are twin prime numbers that occurred in the middle of prime month, and the serial number is 333899[ 18 1]. Any natural number n greater than 1 can be uniquely decomposed into the product of a finite number of prime numbers n = (P _ 1 A 1) * (P _ 2 A2) ... (P _ N An), where p _1< p _ 2 <; ... & ltP_n is a prime number and its power ai is a positive integer.
This decomposition is called the standard decomposition of n.
The content of fundamental theorem of arithmetic consists of two parts: the existence of decomposition and the uniqueness of decomposition (that is, the way in which a positive integer is decomposed into a prime product is unique regardless of the arrangement order).
Fundamental theorem of arithmetic is a basic theorem in elementary number theory, and it is also the logical support and starting point of many other theorems.
This theorem can be extended to more general commutative algebra and algebraic number theory. Gauss proved that the complex integer ring Z[i] also has a unique decomposition theorem. The concepts of unique decomposition of whole rings and Euclidean whole rings are also summarized. More generally, there is Dai Dejin's ideal decomposition theorem. Arithmetic progression is a kind of sequence. In arithmetic progression, the difference between any two adjacent terms is equal. This difference is called tolerance. Similar to 7, 37, 67, 97, 127, 157. Such a series composed of prime numbers is called arithmetic prime number series. In 2004, Green and Tao Zhexuan proved the existence of an arbitrarily long prime arithmetic progression. On April 18, 2004, they announced that they had proved that "there is a prime arithmetic progression with any length", that is, for any value k, there are k prime numbers in arithmetic progression. For example, K=3, there are prime sequences 3, 5, 7 (both of which are 2)...k = 10, there are prime sequences199,409,619,829, 1039,/kloc.