For example, only if 23 is divided by itself (that is, 23) and then by 1, the number obtained is an integer, and it is a prime number because it cannot be divided by other numbers.
2. Composite number: refers to the number in the natural number that can be divisible by other numbers (except 0) except 1 and itself.
For example, 4 can be divisible by itself (that is, 4) and 1, or it can be divisible by 2 to get an integer, so it is a composite number, and 4 is also the smallest composite number.
Prime number:
A prime number is defined as a natural number greater than 1, and there are no other factors except 1 and itself.
Property: The number of prime numbers is infinite.
Prime number theorem:
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.
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.
5. Even numbers must be written as a prime number plus a composite number consisting of at most five factors. Later, someone abbreviated this result to (1+5).
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)?
Nature:
Prime numbers have many unique properties:
The (1) prime p has only two divisors: 1 and p.
(2) Basic theorem of elementary mathematics: Any natural number greater than 1 is either a prime number itself or can be decomposed into the product of several prime numbers, and this decomposition is unique.
(3) The number of prime numbers is infinite.
(4) The number formula π(n) of prime numbers is an irreducible function.
(5) If n is a positive integer, between n 2 and (n+ 1) 2? There is at least one prime number between.
(6) If n is a positive integer greater than or equal to 2, it is between n and n! There is at least one prime number between.
(7) If the prime number p is not greater than n (n >; =4), then p> is not applicable.
(8) Among all prime numbers greater than 10, the unit number is only 1, 3, 7, 9.
Composite number:
1, all even numbers greater than 2 are composite numbers.
2. In all odd numbers greater than 5, all numbers with 5 are composite numbers.
3. Except 0, all natural numbers whose unit is 0 are composite numbers.
4. All natural numbers with units of 4, 6 and 8 are composite numbers.
5. The smallest (even) complex number is 4, and the smallest odd complex number is 9.
6. Every composite number can be written as the unique form of the product of prime numbers, that is, the factorization of prime factors. (fundamental theorem of arithmetic)
7. For any composite number greater than 5 (Wilson theorem): (p- 1)! =- 1(modp)