Proof of infinity of prime numbers
The number of prime numbers is infinite. The most classic proof was proved by Euclid and recorded in his Elements of Geometry. It uses the commonly used proof method: reduction to absurdity. The specific proof is as follows:
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.
Used to calculate the number of prime numbers in a certain range.
Although the whole prime number is infinite, some people will ask, "How many prime numbers are there below 100000?" "What is the probability that the random number of 100 is a prime number?" . The prime number theorem can answer this question.
Editing this famous problem Goldbach conjecture
Goldbach put forward the following conjecture in his letter to Euler in 1742: any integer greater than 2 can be written as the sum of three prime numbers. Because the convention that "1 is also a prime number" is no longer used in mathematics, the modern statement of the original conjecture is that any integer greater than 5 can be written as the sum of three prime numbers. Euler also put forward another equivalent version in his defense, that is, any even number greater than 2 can be written as the sum of two prime numbers. Today's popular conjecture is said to be Euler's version. Any sufficiently large even number can be expressed as the sum of a number with no more than one prime factor and a number with no more than b prime factors, and the proposition is called "a+b". 1966 Chen Jingrun proved that "1+2" holds, that is, "any sufficiently large even number can be expressed as the sum of two prime numbers, or the sum of a prime number and a semi-prime number". The common conjecture today is the Euler version, that is, any even number greater than 2 can be written as the sum of two prime numbers, which is also called "strong Goldbach conjecture" or "Goldbach conjecture about even numbers".
It can be inferred from Goldbach's conjecture about even numbers that any odd number greater than 7 can be written as the sum of three prime numbers. The latter is called "weak Goldbach conjecture" or "Goldbach conjecture on odd numbers".
If the Goldbach conjecture about even numbers is right, then the Goldbach conjecture about odd numbers will also be right. The weak Goldbach conjecture has not been completely solved, but in 1937, vinogradov, a mathematician of the former Soviet Union, proved that an odd prime number large enough can be written as the sum of three prime numbers, which is also called Goldbach -V Noguera Dov theorem or triple prime number theorem. Mathematicians believe that the weak Goldbach conjecture has been basically solved.
Riemann hypothesis
Riemann conjecture is a conjecture about the zero distribution of Riemann zeta function zeta (s) put forward by mathematician Bernhard Riemann (1826- 1866) in 1859. Hilbert, a German mathematician, listed 23 mathematical problems, among which Riemann hypothesis was included in the eighth problem. There is no simple rule for the distribution of prime numbers in natural numbers. The frequency of Riemann discovering prime numbers is closely related to Riemann zeta function. Riemann conjecture puts forward the nontrivial zero of Riemann zeta function zeta (s) (here, s is not the value of points -2, -4, -6, etc.). ) is 1/2. That is to say, all nontrivial zeros should lie on the straight line 1/2+ti ("critical line"). T is a real number and I is the basic unit of imaginary number. So far, no one has given a convincing and reasonable proof of Riemann conjecture.
In the study of Riemann conjecture, mathematicians call the straight line Re(s)= 1/2 on the complex plane the critical line. Using this term, Riemann conjecture can also be expressed as: all nontrivial zeros of Riemann zeta function are located on the critical line.
Riemann conjecture was put forward by Riemann in 1859. In the process of proving the prime number theorem, Riemann put forward the conclusion that all zeros of Zeta function are on the straight line Res(s) = 1/2. He gave up after his proof failed, because it had little effect on his proof of the prime number theorem. But this problem has not been solved so far, and even a simpler guess than this assumption has not been proved. Many problems in function theory and analytic number theory depend on Riemann hypothesis. The generalized Riemann hypothesis in algebraic number theory has far-reaching influence. If we can prove the Riemann hypothesis, we can solve many problems.
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.
The prime numbers within 100 are 2,3,5,7, 1 1 3, 17,19,23,29,31.
Fermat number 2 (2 n)+ 1
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. Sixty-seven years after Fermat's death, Euler, a 25-year-old Swiss mathematician, proved that F5 was a composite number.
Later, mathematicians never found out which Fn values are prime numbers, all of which are composite numbers. At present, due to the large square, there are few proofs. Now mathematicians get the maximum value of Fn: n= 1495, and its digits are as high as 10 10584. Of course, although it is big, it is not a prime number.
mersenne prime
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.
At present, the maximum mersenne prime discovered by mathematicians is 243112609-1.
Edit this paragraph related theorem prime number theorem
The prime number theorem describes the general distribution of prime numbers. The appearance of prime numbers has been puzzling mathematicians. One by one, the appearance of prime numbers in positive integers is irregular. But in general, the number of prime numbers is regular. For the positive real number X, π(x) is defined as the number of prime numbers not greater than X. Mathematicians have found some functions to estimate the growth of π(x). The following is the first such estimate. π(x)≈x/ln x where ln x is the natural logarithm of x, the above formula indicates that when x approaches∞, the ratio of π(x) to x/ln x approaches 1 (note: this result was discovered by Gauss). But this does not mean that their values are close with the increase of X. Here is a good estimate of π(x): π (x) = li (x)+o (xe (-(ln x) (1/2)15), when x is close to ∞. Where Li(x) = ∫(dt/ln x2, x), and the second term on the right side of the relation is error estimation.
The prime number theorem can give the asymptotic estimation of the nth prime number p(n): p (n) ~ n/ln n, and also give the probability of extracting a prime number from an integer. Randomly choose a natural number not greater than n, and the probability that it is a prime number is about1/ln n. The formula of this theorem was put forward by the French mathematician Legendre in 1798. 1896, French mathematician Jacques Hadamard and Belgian mathematician Charles Jean de la Vallé e-Pu Sang gave independent proofs respectively. It is proved that complex analysis, especially Riemann zeta function, is used. Because Riemann zeta function is closely related to π(x), Riemann conjecture about Riemann zeta function is very important to number theory. Once the conjecture is proved, the error estimation of the prime number theorem can be greatly improved. 190 1 year, Swedish mathematician Helge von Koch proved that if Riemann conjecture holds, the estimation of the error term in the above relation can be improved to π (x) = Li (x)+O (x (1/2) LNX), but the constant of the big o term is unknown. Some elementary proofs of the prime number theorem only need the method of number theory. The first elementary proof was obtained in 1949 by Hungarian mathematician Paul Edith ("Erdos" or "Erdoshi") and Norwegian mathematician Atree Silberg. Before this, some mathematicians did not believe that elementary proofs could be found without the help of difficult mathematics. British mathematician Hardy said that the prime number theorem must be proved by complex analysis, which shows the "depth" of the theorem result. He believes that only using real numbers is not enough to solve some problems, and complex numbers must be introduced to solve them. This is based on the feeling that some methods are more advanced and powerful than others, and the elementary proof of prime number theorem shakes this argument. Selberg-Edith's proof just shows that seemingly elementary combinatorial mathematics can also be very powerful. However, it should be pointed out that although this kind of elementary proof only uses elementary methods, it is even more difficult than using complex analysis.
fundamental theorem of arithmetic
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 series of prime numbers
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, 107, 137, 167, 197. 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.
reference data
1. The results of Green and Tao Zhexuan-proved the existence of prime numbers of arbitrary length. Authors: Green, B. and Tao, T; Thesis title: Prime numbers contain arbitrary length and arithmetic series; Date of submission: April 9, 2004; Date of acceptance: September 2005 12; Publish a magazine: yearbook