(Example 1),,,, That is to say, any number is composed of prime numbers.
(Example 2) 2 = (1× 2), 3, 5, 7, 1 1 … are all prime numbers, while 4, 6 and 8 are not prime numbers (because there are at least factors of 2).
Because of the singularity of prime number itself, people can't grasp the law of its appearance, grasp its characteristics, or even know its actual distribution. Simply put, given a positive integer, you can't know whether it is a prime number or not. Even if you have exhausted all methods to prove that it can't be a prime number, you can't decompose it. For example: 211-1= 2047 can be decomposed into .267-1. It is said that it took American mathematician Frank Nelon Cole more than three years to discover it. Naturally, the "computer age" is still coming, only by infinite patience and perseverance. Plus a training that takes a long time to compare and calculate. But using a computer seems to be no better. The number of words has increased, and the difficulty remains the same. 193 1 year D.H. Lehmar proved that 2257- 1 is a large composite number. Big! Not bad. Equal to 23 1 584,178,474,632,390,847, 14 1 970,017,375,8/kloc-.
539,969,33 1,28 1, 128,078,9 15, 168,0 15,826,259,279,87 1
A large number of 78-bit numbers, so far no one or computer can decompose them!
Therefore, although it may not be useful to know whether a number is a prime number, it is still very interesting, at least it will cause many methodological problems in the process of finding it.
Characteristics of prime numbers
The prime number of 1 must be odd except 2 (in other words, 2 is the smallest prime number and the only even number).
2'1'is not a prime number.
3 fundamental theorem of arithmetic: Any integer greater than 1 must be decomposed into the product of prime factors, and the representation method is unique.
The number of prime numbers and its solution
1 Euclid proves that "there must be an infinite number of prime numbers"
2 "eratostenes" filter sleeve
If you want a prime number from 2 to n, just check whether n is divisible by a prime number not greater than. To judge whether 3 13 is a prime number, we only need to check whether 3 13 can be divisible by a prime number less than or equal to 17.
Are there any special types of prime numbers?
Mersenne prime: If it is a prime number, it is called (but a prime number is not necessarily a prime number,
For example, there are 38 non-prime numbers, such as 3,7,31,127 and so on. , still searching ...
Fermat prime number: when n=0 to 4, type such as. (But prime numbers are not necessarily types, for example.
If n=5, it is a non-prime number. )
The note type is fermat number, and the Fermat prime number is only 3,5,17,257,65537.
Can you use a formula to represent all prime numbers?
(1) Euler:: When x=0, 1, 2…40, you can get 4 1 prime numbers.
(1) Lejeune:: When x=0, 1, 2…28, 29 prime numbers can be obtained.
When x=0, 1, 2…79, 80 prime numbers can be obtained.
When x= 1, 2… 1 1000, we can get 1 1000 prime numbers.
However, no polynomial can represent all prime numbers.
Why are you looking for prime numbers?
"Since there are infinitely many prime numbers, why do mathematicians spend so much effort to find bigger prime numbers?"
Simply put, mathematicians are just like ordinary people. "Do you have a hobby of collecting things" and "Do you like to get a place in the competition" are all reasons. To answer this question, we can explain it from several directions.
One, this is tradition!
Euclid began this pursuit in 300 BC! He mentioned the concept of perfect number in Geometric Prime Numbers, which was related to Masini Prime Numbers, and opened the door to research. Later, great mathematicians such as Fermat, Euler, Masini and Descartes devoted themselves to this pursuit. In the process of searching for large prime numbers, it is of great help to the basic number theory, so this tradition of searching is worth continuing ~
Second, its added value!
Because of the political purpose of the United States, there is a pioneering work to send people to the moon. But the pursuit of big prime numbers, such as Masini prime numbers, has a lasting impact on society. Its added value lies in constantly promoting the progress of science and technology and the research and development of useful things and materials in people's daily life, and improving education construction to make life more productive. In the process of finding and recording Masini prime numbers, teachers can guide students to conduct research, which can enable students to apply the spirit of research to their work and projects.
Third, everyone likes beautiful and rare things!
As mentioned above, after Euclid began this pursuit, it is so rare (there are more than 30 known species, and it is still looking for it). Not only that, it is also beautiful; What is "beauty" in mathematics? For example, people want to prove that it is short and clear, and it can integrate old knowledge and let you know new things! The form and proof of Masini prime numbers meet the above requirements.
Fourth, great glory!
Why do athletes keep pursuing higher, faster and farther? Do they want to use these skills in their work? No, they are all eager to compete and win! Steep cliffs and towering peaks have irresistible charm for people who like rock climbing and mountain climbing, as well as the exploration of mathematics. It's the same feeling when watching an unimaginable huge number turn out to be a prime number, so the desire to continue looking for the next one can't be described in words.
Of course, people need to be pragmatic, but they also need curiosity and the spirit of continuous efforts and progress.
Fifth, computer testing!
After the invention of computers, people can find Marcy prime numbers by computer calculation, because it takes more than one billion calculations to test a known prime number (computers are of course fast). This is a good time to test the stability of the computer. When Thomas accurately calculated the twin prime number constant, Intel's Pentium processor was found to have an error.
Sixth, understand the distribution of prime numbers!
Although mathematics is not an experimental science, now we will test our guesses with examples. When there are more and more examples, we will know more about the facts, and the distribution of prime numbers is like this. For example, Gauss guessed the prime number theorem after reading the prime number table, and Hadamard and Pouusin proved in 1896 respectively:
Prime numbers are part of natural numbers. Interestingly, there are as many natural numbers as there are, and there are infinitely many. More than two thousand years ago, ancient Greek mathematicians proved this in theory. However, prime numbers seem to be much less than natural numbers. According to statistics, there are 1 68 prime numbers between 1000. Between 1000 and 2000, there are 135 prime numbers; Between 2000 and 3000, there are 127 prime numbers; Between 3000 and 4000, there are only 120 prime numbers, and the later, the rarer the prime numbers are. So, how to find prime numbers from natural numbers? In the 3rd century BC, the ancient Greek mathematician Eratosthenes invented a very interesting method. Eratosthenes often writes tables on boards with Bai La painted on them. When he meets the numbers that need to be crossed out, he will find them. As the composite numbers are crossed out one by one, the chessboard becomes riddled with holes, just like a magic sieve, which sifts out the composite numbers and leaves the prime numbers. Therefore, people call this method of finding prime numbers "Elatoseni screening method".
1. We list the natural numbers from 1 to 100 into one hundred tables in sequence (as shown in the following table).
2. Cross out 1 first, because 1 is neither prime nor composite.
3. The next number is 2, which is the smallest prime number and should be kept. However, the multiple of 2 must not be a prime number and should be completely crossed out. That is, starting from 2, every number of 1 is crossed out.
Among the remaining numbers, 3 is the first one that has not been crossed out. This is a prime number and should be kept. However, multiples of 3 must not be prime numbers and should be completely crossed out. That is, starting from 3, cross out 1 every 2 numbers.
5. Among the remaining numbers, 4 was crossed out, and the remaining number 5 became the first number that was not crossed out. This is a prime number and should be kept. But the multiple of 5 must not be a prime number, and it must be completely crossed out; That is, starting from 5, cross out 1 every 4 numbers.
6. Imitate step 1 ~ 5, and continue to row. The last remaining number on the table is the prime number between 1~ 100.
Elatoseni screening method
This method is the oldest method to find prime numbers in the world. Its principle is simple and it is convenient to use. Now, through the improved Elatoseni screening method, mathematicians have screened out all prime numbers within 654.38+0 billion. It is said that Greece and China began to ask how to find prime numbers from the Zhou Dynasty. The following are some preliminary questions.
Prime numbers are infinite. This has long been proved. Because if P 1 = 2, P2 = 3, and PN is the first n prime numbers, then the new number must be divided by a new prime number that is not equal to any of p 1=2, p2=3, PN and pn, so pn+ 1 exists. and
For example,
But 3003 1=59 x 509
It is proved that it is not necessarily a prime number.
think
Is there an infinite number of prime numbers in f(n) or an infinite number of composite numbers in f(p)?
How to prove that n is prime?
The traditional "screening method" is to verify any possible factor of the number n and simplify it; Only filter all prime numbers less than. That is to say, if n is a composite number, there must be a prime factor less than, such as 3, 5, 7, 1 1, 13 and so on. At present, there are piecemeal methods to check prime numbers, but there is still no perfect solution.
Fermat conjecture
/kloc-in the 7th century, there was a French lawyer named Fermat (Fermat, 160 1- 1665). He likes mathematics very much and often studies advanced mathematics problems in his spare time. As a result, he achieved great success and was called "the king of amateur mathematicians". When Fermat studied mathematics, he didn't like to do it. With rich imagination and profound insight, he put forward a series of important mathematical conjectures, which deeply influenced the development of mathematics. His Fermat's Last Theorem has attracted countless mathematicians for hundreds of years, and it was not until 1994 that it was proved by wiles of Princeton University in the United States.
In 1640, he proposed a formula: "2+ 1". He checked the situation that n is equal to 1 ratio 4, and found that they are all prime numbers (as shown in the following table), and directly guessed that as long as n is a natural number, the formula must be a prime number. "
n
2+ 1
1
2+ 1=5 (prime number)
2
2+ 1= 17 (prime number)
three
2+ 1=257 (prime number)
four
2+ 1=65537 (prime number)
1. Fermat's favorite branch of mathematics is number theory. He deeply studied the properties of prime numbers, and he found that? C