First, the prime number theorem
Two nonzero natural numbers whose common factor is only 1 are called prime numbers; For example: 2 and 3, the common factor is only 1, which is a prime number;
2. Some positive integers whose greatest common factor is only 1 are called prime numbers;
3. Two different prime numbers are prime numbers;
4. 1 is coprime with any natural number. Two different prime numbers are coprime. A prime number and a composite number are coprime when they are not multiples. The coprime of two complex numbers without the same prime factor;
5. Any two adjacent numbers are coprime;
6. The probability that two positive integers are coprime (the greatest common divisor is 1) is 6/π 2.
Second, the use of expressions.
The "two numbers" mentioned here refer to all natural numbers except 0. "The common factor is only 1", which cannot be mistaken as "no common factor." There are two different situations in which three or more natural numbers are coprime: one is that these natural numbers that become coprime are pairwise coprime. For example, 2, 3 and 5.
The other one is not in pairs. For example, 6, 8 and 9. When two integers (positive integers) (n) have no common divisor except 1, they are called prime numbers. The probability of a prime number is 6/π 2. When two prime numbers are multiplied, the number obtained is not necessarily a composite number.
Because one and any non-zero natural number are coprime, the product obtained by multiplying one by any non-zero natural number is not necessarily a composite number. For example, 1 and 17 are coprime, and 1× 17= 17 is not a composite number.
Method of judging prime numbers
First, the concept judgment method
Two numbers whose common divisor is only 1 are called prime numbers. According to the concept of coprime number, we can judge whether a group of numbers are coprime. For example, the common divisor of 9 and 1 1 is only 1, and they are prime numbers.
Second, the legal judgment method
According to the definition of prime numbers, some laws can be summarized, which can be used to quickly judge whether a group of numbers are prime numbers.
Third, decomposition judgment method
If both numbers are composite numbers, we can first decompose the two numbers into prime factors respectively, and then see whether the two numbers contain the same prime factors. If not, these two numbers are prime numbers. For example: 130 and 23 1, first decompose them into prime factors:130 = 2× 5×13,231= 3× 7×1/kloc-0. After decomposition, it is found that they do not have the same prime factor, so 130 and 23 1 are prime numbers.
Fourth, the difference judgment method
If the difference between two numbers is not big, we can find their difference first, and then see if the difference and the smaller number are coprime. If it is coprime, then the original two numbers must be coprime. For example: 194 and 20 1, first find their difference, 20 1-kloc-0/94 = 7, because 7 and 194 are coprime, then 194 and 20/kloc-.
Five, business judgment method
Divide a large number by a decimal. If the remainder is coprime with the smaller number, the original two numbers are coprime. For example: 3 17 and 52, 3 17 ÷ 52 = 6...5, because the remainders 5 and 52 are coprime, then 3 17 and 52 are coprime numbers.