Basic knowledge of Number Theory

Basic knowledge of Number Theory

This article briefly introduces the Integer Set z = {..., -2 ,...} And natural number set n = {0, 1, 2 ,...} The most basic concept of number theory.

Division and appointment

The concept that an integer can be divisible by another integer is a central concept in number theory. The mark d | A (read as "D division A") means that for an integer k, There Is A = KD. 0 can be divisible by each integer. If a> 0 and d | A, | d | ≤ | A |. If d | A, we can also say that A is a multiple of D. If a cannot be divisible by D, write DFA.

If d | A and D ≥ 0, then we say D is the approximate number of. Note: d | A when and only when (-d) | A, so defining an appointment number as a non-negative integer will not lose its universality, as long as you understand that the corresponding negative number of any appointment number of a can also divide. The minimum and maximum values of an integer a are 1 | A |. For example, the approximate numbers of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Each integer A can be divisible by its ordinary approx. 1 and. The extraordinary divisor of A is also called the factor of. For example, 20 has 2, 4, 5, and 10.

Prime Number and Union number

For an integer a> 1, if it has only ordinary approx. 1 and A, we call a as a prime number (or prime number ). Prime numbers have many special properties and play an important role in number theory. In order, the following is a sequence of small prime numbers:

, 59 ,...

An integer a> 1 that is not a prime number is called a union. For example, 39 is a combination because there are 3 | 39. An integer 1 is called a base. It is neither a prime nor a combination. Similarly, integer 0 is neither a prime nor a combination of all negative integers.

Theorem 1

There are infinite prime numbers.

Proof:

Suppose there are only N prime numbers, which are arranged in ascending order as P1, P2 ,..., pn, then x = (P1 · P2 ·... · Pn) + 1 obviously cannot be P1, P2 ,..., any prime number in PN is divisible, So X is also a prime number, which is in conflict with only N prime numbers, so prime numbers are infinitely many.

This proof was first from Aristotle. It was very beautiful and a classic application of the anti-Evidence method. This proof was called by Euler "A proof from God directly", and mathematicians of history spoke highly of it.

Division theorem, remainder and same Modulus

Given an integer N, All integers can be classified as integers that are multiples of N and integers that are not multiples of N. For integers that are not multiples of N, we can divide them by the remainder of N. Most of the theory of number theory is based on the above division. The following theorem is the basis for such division.

Theorem 2 (Division theorem)

For any integer a and any positive integer N, there is a unique integer Q and R, satisfying 0 <R≤NAndA = qn + R.

This theorem is one of the basic theorems of integers.

The Q = Operator A/n operator is the division operator (operator x operator represents the floor symbol floor, that is, the maximum integer smaller than or equal to X ). ValueR=AMoDNIt is called the remainder of division. We haveN|AWhen and only whenAMoDN= O, and has the following formula:

(1)

Or

(2)

When we define the concept of dividing an integer by the remainder of another integer, we can easily give a special note that represents the same remainder. If (AMoDN) = (BMoDN).ABytesB(ModN), And saidAAndBPeer modeNIs equal. In other words, whenAAndBDividedNWhen there is the same remainderABytesB(ModN). It is equivalent,ABytesB(ModN) When and only whenN| (B-A). IfAAndBPeer modeNIf not equal, writeATB(ModN). For example, 61 Then 6 (mod 11), similarly,-13 then 22 then 2 (mod 5 ).

Based on Integer ModulusNThe remainder can be an integer.N. ModuleN Integer contained in the equivalence classAIs:

For example, [3] 7 = {..., -11,-4, 3, 10, 17 ,...}, This set also has other note methods [-4] 7 and [10] 7.A[B]N. It is equivalentABytesB(ModN). The set of all such equivalence classes is:

(3)

We often see definitions

(4)

If 0 is used to represent [0] n, 1 is used to represent [1] n. And so on. Each class is represented by its smallest non-negative element. The two definitions (3) and (4) are equivalent. However, we must remember the corresponding equivalence classes. For example, the element-1 mentioned in Zn refers to [n-1] n, because-1 = N-1 (mod N ).

Common and maximum common

If D is an appointment of A and is also an appointment of B, D is the appointment of A and B. For example, the common numbers of 24 and 30 are 1, 2, 3, and 6. Note that 1 is the approximate number of any two integers.

An important attribute of a common appointment is:

(5)

Generally, for any integer x and y, we have

(6)

Similarly, if A | B, or | A |≤| B |, or B = O, which indicates:

(7)

The maximum common divisor of two integers, A and B, which are not at the same time 0, is expressed as gcd (A, B ). For example, gcd (24, 30) = 6, gcd (5, 7) = 1, gcd () = 9. If both A and B are 0, gcd (A, B) is an integer between 1 and min (| A |, | B |. We define gcd (O, 0) = 0, which is essential to make the general nature of GCD functions (the following formula (11) Universally correct.

The following are the basic properties of GCD functions:

	(8)
	(9)
	(10)
	(11)
	(12)

Theorem 3

If A and B are not any Integers of 0, then gcd (A, B) is a linear combination of A and B. {ax + by: X, the smallest positive element in y, z.

Proof:

Set S to the smallest positive element in the linear combination of A and B, and for an X, Y, Z, S = AX + by, set q = ë A/S distinct, formula (2) Description

Therefore, a mod S is also a linear combination of A and B. But since a mod S <s, we have a mod S = O, because s is the smallest positive number that satisfies such a linear combination. So there is S | A, and it is similar to S | B. Therefore, S is the common relationship between A and B, so gcd (A, B) is ≥s. Because gcd (A, B) can be divisible by A and B at the same time, and S is a linear combination of A and B, we can see from formula (6) that gcd (A, B) | S. But from gcd (a, B) | S and S> O, we can see that gcd (A, B) is less than or equal to S. It has been proved above that gcd (A, B) is ≥s, so we get gcd (a, B) = S. Therefore, we can obtain the maximum public approx. S is a and B.

Inference 4

For any integer A and B, if d | A and D | B, d | gcd (A, B ).

Proof:

According to Theorem 3, gcd (A, B) is a linear combination of A and B, so this inference can be inferred from formula (6.

Inference 5

All integersA AndBAnd any non-negative integerN, Gcd (An,BN) =NGcd (A, B).

Proof:

If n = 0, this inference is obvious. If n = 0, gcd (an, bn) is the minimum positive element in the {anx + BNY} set, that is, n times the minimum positive element in the {ax + by} set.

Inference 6

For all positive integers n, A, and B, if n | AB and gcd (A, n) = 1, then n | B.

Proof:

(Omitted)

Quantity

If two integers A and B have only a public factor of 1, that is, if gcd (a, B) = 1, then A and B are called the mutual prime numbers. For example, 8 and 15 are mass numbers, because the approximate number of 8 is 1, 2, 4, 8, and the approximate number of 15 is 1, 3, 5, 15. The following theorem indicates that if each number in two integers is a prime number with an integer p, their product and P are mutually prime numbers.

Theorem 7

For any integer A, B, and P, if gcd (A, P) = 1 and gcd (B, P) = 1, then gcd (AB, P) = 1.

Proof:

According to Theorem 3, integers x, y, x', And y' are satisfied.

Ax + py = 1

BX '+ py' = 1

Multiply the two sides of the preceding equations. We have

AB (xx') + P (ybx '+ y' AX + pyy') = 1

Because 1 is a positive linear combination of AB and P, we can use Theorem 3 to prove the conclusion.

For integers N1, N2 ,..., NK. If gcd (Ni, NJ) = 1 exists for any I =j, it is an integer N1, N2 ,..., NK mutual quality.

Unique Factorization

The following conclusions describe a basic but important fact about the division of prime numbers.

Theorem 8

For all prime numbers P and All integers A, B, if p | AB, PLA or p | B.

Proof:

To introduce contradictions, we assume p | AB, but PfA and pfb. Therefore, gcd (A, P) = 1 and gcd (B, P) = 1, because the approximate number of P is only 1 and P. It is assumed that P cannot be divisible by A or B. According to theorem 7, we can see that gcd (AB, P) = 1; then, assuming p | AB, we can know that gcd (p, AB) = P, which leads to a conflict.

It can be inferred from theorem 8 that the factorization formula of an integer into prime numbers is unique.

Theorem 9 (unique prime factor decomposition)

Any integer a can and can only be written into the following product form

Where pi is the I-th prime number in the natural number, P1 <P2 <... <PR, and ei is a non-negative integer (note that EI can be 0 ).

Proof:

(Omitted)

For example, hundreds of thousands can be uniquely divided into 24, 31, 53.

this theorem is very important . in computational theory, many important theorems can be proved based on this theorem, this theorem actually gives a one-to-one correspondence between Z and z *. In other words, any integer a can use a group of Integers (E1, E2 ,..., ER), and vice versa, where a and (E1, E2 ,..., Er) the formula that satisfies theorem 9. For example, 6000 can be expressed as a group of Integers (6000, 3), because = 24 · 31 · 53. From another perspective, this also provides a large integer compression method. Unfortunately, this compression method is not feasible due to the time-consuming decomposition :-(. However, in the proof of many theorem (especially in the theorem of Computational Theory, formal language, and mathematical logic), this method can be used to represent a string of integers with a unique integer. For example, in a Turing machine, the input is an integer with an indefinite length. After some conversion, the output is another integer with an indefinite length, we can use Theorem 9 to represent input and output with a unique integer. In this way, the conversion process is treated as a simple function from an integer set to an integer set, it provides convenience for us to study this conversion process theoretically. This technique is used in the proof of geder's uncertainty principle, so this encoding method is also called Godel encoding. For another simple example, if we encode each Chinese Character and use an integer to represent a Chinese character, the English 26 letters are encoded and an integer is used to represent a letter, now we need to translate a sentence into an English sentence for output. Both input and output can be represented by a group of integers, which are encoded by the Corder above, the input and output can be represented by a unique definite integer. The translation process is a function operation. This function is a simple integer function of Z → Z. If you find this function, then we made a machine translation machine. In fact, in the world, all algorithms that can be processed using can be converted to simple integer functions using Godel encoding. to study, this is why the computation theory only studies simple integer functions.