ACM number theory summary

I have been learning number theory intermittently for a while, and I have learned a lot. Now I will make some simple review and summary.

You can't forget what you learned...

 

1. original stock Number:

Concept: a triple (A, B, C), where A, B, and C have no public factor and meet the following requirements: A ^ 2 + B ^ 2 = C ^ 2

First, the number of such original Hooks is infinite, and the constructed conditions are met:

A = S * T, B = (s ^ 2-T ^ 2)/2, c = (s ^ 2 + t ^ 2)/2

S> T> = 1 is an odd number without a public factor!

From the above concepts, you can export any original check array.

 

2. prime number counting (Prime Number Theorem)

Set π (X) to the number of prime numbers in 1 to X.

The highest number theory achievement in the 19th century is the following:

Lim (X-> ∞) {π (X)/(x/ln (x)} = 1

The highest achievement and the highest achievement in number theory !!! Yes !!!

 

3. godbach conjecture (1 + 1)

A large even number (> = 4) can be split into two prime numbers. Although no one can prove it theoretically, based on the computation results of scientists, if an even number cannot be split, the even number must be at least one hundred digits !!

Therefore, in the ACM world (the data volume is usually less than 2 ^ 63), the golden Bach conjecture is true !! So the splitting program will be able to implement

 

4. Promotion of goldebach Conjecture

Any integer greater than or equal to 8 can be split into the sum of four prime numbers.

Proof:

First, 8 = 2 + 2 + 2 + 2, (the sum of the four smallest prime numbers) can no longer find a prime number smaller than 2, so when n is less than 8, it cannot be split into the sum of four prime numbers!

When N is greater than or equal to 8, we can discuss the situation as follows:

(1) N & 1 = 0 (N is an even number), then N can be split into two even numbers and

Then, according to goldebach's conjecture, an even number can be split into two prime numbers. Therefore, N must be split into four prime numbers.

(2) N & 1 = 1 (n is an odd number). N must be split into two even numbers + 1.

Since there is a prime number that is an even number and 2, the odd number must be split as follows: 2 + 3 + Prime Number

.

 

5. Euler's function (Euler's formula)

Euler's function pH (n) indicates the number of all numbers less than N and with N

For example, pH (12) = 4, [1, 5, 7, 11 and 12]

Euler's Formula

A ^ pH (m) = 1 (mod m)

 

6. ferma's Theorem

Ferma's theorem is a special case of Euler's formula.

Because when P is the prime number, pH (p) = p-1 is obvious.

Then the Euler's formula is used to obtain the ferma's theorem.

A ^ (p-1) = 1 (mod P)

P is prime)

 

7. Drawer principle

The drawer principle is actually nonsense. The key lies in its application.

If there are three apples in two drawers, there will be at least two apples in one drawer.

 

8. Drawer principle Application

The drawer principle is just a nonsense, but his application is very powerful.

Assume that there is a positive integer sequence A1, A2, A3, a4 ..... an. It is proved that we can find a continuous sequence and make this sum a multiple of n. The proof of this proposition uses the drawer principle.

We can first construct a sequence Si = A1 + A2 +... AI

Then perform modulo operations on Si separately. If one of them is SK % N = 0, then A1 + A2 +... + AK must be a multiple of N (this is proved)

The following is the opposite of the previous case, that is, the remainder of N for any sk is not 0.

In this case, we can consider the following because Si % n! = 0

Then the range of Si % N must be 1 -- (n-1), so the original sequence Si generates n remainder numbers in the range of 1 -- (n-1), so the drawer principle comes, if n numbers are placed in n-1 boxes, there must be at least two remainder numbers, and the difference between the two sk1 and sk2 must be a multiple of N,

While sk1-sk2 is a continuous sequence, the original proposition is proved.

 

9. Judge n! Can be divisible by M?

The calculation method is to separate m from the prime factor to see if each prime factor of M can be at N! ;

N! The number of X contained in the middle (X is an arbitrary number, but we generally only discuss X as a prime number). The answer to this question is:
N/x + N/(x ^ 2) + N/(x ^ 3 )..... [the multiplication side that is always added to X does not exceed n]. The proof of this theorem is also very simple. I will not go into details here.

Based on the above points, we can calculate whether each prime factor of M is completely included. If one is not included, it cannot be divisible!

 

10. Calculation Method of factor sum

Shenma is called a factor and: The sum of a number is called a factor and...

Well, for example, the 12 factor is 1 + 2 + 3 + 4 + 6 + 12.

The calculation method is to break 12 into the expression of prime factor 2 ^ 2*3.

Then the sum of its factors is: (1 + 2 + 2 ^ 2) * (1 + 3)

It turns out to be difficult to write. The general idea is Newton's binary...

 

11. Judge the parity of the combination number C (n, m)

I don't know how to prove it.

If N & M = M is an odd number, the opposite is an even number.

 

I will summarize it here.

In the past, freshman year also summarized a similar article, but at that time it summarized a bit about Euclidean algorithms and so on.

-- Bingshen