From this point of view, integers can be divided into two types, one is called prime number and the other is called composite number. Some people think that the number 1 should not be called prime number. ) the famous gauss "unique decomposition theorem" says that any integer. It can be written as the product of a series of prime numbers.
A composite number, also called a composite number, is a positive integer that satisfies one of the following (equivalent) conditions:
1. is the product of two integers greater than 1;
2. There is a factor)； greater than 1 and less than itself;
3. There are at least three factors (factors);
4. It is neither 1 nor a prime number (prime number);
5. Non-prime numbers with at least one prime factor.
The following are conclusions about complex numbers and some special complex numbers:
A composite number has an odd number of factors (factors) if and only if it is a complete square number.
A number with 1, only 1 and its own two divisors is called a prime number. (For example, 2 ÷ 1 = 2,2 ÷ 2 =1,so the divisor of 2 is only1and itself 2,2 is a prime number. )
2. Besides 1 and its two divisors, there are other divisors called composite numbers. (For example, 4 ÷1= 4,4 ÷ 2 = 2,4 ÷ 4 =1. Obviously, the divisor of 4 includes the divisor 2 besides the two divisors of 1 and itself 4, so 4 is a composite number. )
3. 1 is neither a prime number nor a composite number. Because its divisor has only the divisor of 1
The number of prime numbers is infinite. There is a classic proof in Euclid's Elements of Geometry. It uses a common proof method: reduction to absurdity. The concrete proof is as follows: Suppose there are only a limited number of prime numbers, which are arranged as p 1, p2, ..., pn from small to large, and let n = P 1× P2×...× PN, then is it a prime number? If it is a prime number, it should be greater than p 1, p2, ..., pn, so it is not in those hypothetical prime numbers.
1, if it 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 it is impossible to 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. So the original assumption doesn't hold water. In other words, there are infinitely many prime numbers.
2. Other mathematicians have given some different proofs. Euler proved by Riemann function that the sum of reciprocal of all prime numbers is divergent, Ernst Cuomo proved more succinctly, and harry Furstenberg proved by topology.
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.
1, there must be at least one prime number between a number greater than 1 and its twice (that is, within the interval (a, 2a)).
2. There is a prime arithmetic progression of any length. ?
3. An even number can be written as the sum of two composite numbers, and each composite number has at most 9 prime factors. (Norwegian mathematician Brown, 1920)
4. Even numbers must be written as prime numbers plus composite numbers, in which the number of factors of composite numbers has an upper bound. (Renee, 1948)
5. Even numbers must be written as a prime number plus a composite number consisting of at most five factors. Later, some people called this result (1+5) (Pan Chengdong, China, 1968).
6. A sufficiently large even number must be written as a prime number plus a composite number consisting of at most two prime factors. Short for (1+2)
One method of composite number is to calculate the number of prime factors. The sum of two prime factors is called semi-prime, and the sum of three prime factors is called wedge number. In some applications, composite numbers can also be divided into composite numbers of odd prime factors and composite numbers of even prime factors. For the latter, (where μ is Mobius function and'' x'' is half the number of prime factors), while the former is?
Note that for prime numbers, this function returns-1, and. For the number "n" with one or more repeated prime factors,
Another way to classify composite numbers is to calculate the number of their factors. All composite numbers have at least three factors. The square of a prime number whose factor is. If a number has more factors than its small integer, it is called a high composite number. In addition, the number of factors of a complete square number is odd, and other composite numbers are even.
Composite numbers can be divided into odd and even numbers, basic composite numbers (divisible by 2 or 3), negative composite numbers (6N- 1) and positive composite numbers (6N+ 1), two-factor composite numbers and multi-factor composite numbers.