Number Theory 4.3 prime examples

There are infinite prime numbers. Euclidean recursion proves:

Pn = P1P2... Pn-1 + 1, when n ≥ 1

The n-1-1 prime number can not be divided into Pn, because each can be divided into Pn-1.

For example, the number of 2p-1 (P is a prime number) is called Mersenne numbers.

If this number is also a prime number, it is called the prime number of Mason.

If n is a union, 2n-1 cannot be a prime number.

Proof: 2 km-1 = (2 m-1) (2 m (k-1) + 2 m (K-2) +... + 1)

But when P is a prime number, 2 P-1 is not always a prime number.

For example, the minimum non-merceon number is 211-1 = 2047 = 23*89.

Http://acm.zju.edu.cn/show_problem.php? PID = 1, 2400

 

There is an approximate formula, the K prime number PK ≈ K lnk

This means that when K → ∞, PK/K lnk → 1

Similarly, π (x) can be introduced to indicate the number of prime numbers not greater than X, π (x) ≈ x/lnx

When N or X tends to be infinite, more precise estimation functions are available.

Finally, I talked about the eratoshenes screening method. This is too simple and I don't want to worry about it ~