Prove that there are infinite primes / primes

Before we prove that there are infinite primes, let's understand some basic theorems ：

Prime or prime number ： If more than 1 The integer of p All the positive factors of are p and 1, They are called prime numbers or prime numbers （prime）; Otherwise, it is called Sum （composite number）.

Be careful ：1 It's neither prime nor sum .

Basic theorem of arithmetic ： Any one is greater than 1 The natural number of $N$, If $N$ Not prime , that $N$ It can be uniquely decomposed into the product of finite prime numbers $N=P_1^{a_1}\ast P_2^{a_2}\ast P_3^{a_3}\ast .......\ast P_n^{a_n}$, here $P_1<P_2<P_3......<P_n$ All are prime numbers , And the index $a_i$ It's a positive integer. . Such decomposition is called $N$ The standard decomposition of . for example $24=2^3\ast 3$, 2 and 3 Are prime numbers or prime numbers

Let's give Euclid in The elements Li proved the infinity of prime numbers by using the method of contradiction .

1. First, suppose there is a maximum prime number $P$.
2. Then from the 2 To $P$ Multiply all primes between and then add 1：$$N＝2\ast 3\ast 5\ast 7\ast 11\ast .......\ast P+1$$ So you get $N$, $N$ It's a composite number . among $N>P$.
3. According to the basic theorem of arithmetic , There must be a prime number $P_i$ Divisibility $N$, namely $Nmod{P}_i==0$, because $(N-1)mod{P}_i=0$, Then there must be $1mod{P}_i=0$, because $P_i$ The minimum is 2, There is no such thing $P_i$, therefore N It's better than P Larger primes , This contradicts the assumption , That is to prove that there are infinite primes .