Chapter 20      Elementary Number Theory


    This chapter briefly introduces the basic knowledge of elementary number theory . It is divided into six sections . The first five sections discuss the properties of integers and the method of division , continuous fractions and Fibonacci sequences , congruence and Sun Tzu's theorem , and introduce several important The number-theoretic functions and Mobius transforms of algebraic numbers are listed , and the discrimination methods of several types of irreducible polynomials are listed . The last section gives a brief introduction to the basic concepts and properties of algebraic numbers .


§ 1 Integer     


    [ Integer part and fractional part ]   Let a be a real number , the largest integer not exceeding a is called the integer part of α , denoted as . And called the fractional part of a .

    For example , , etc. 

    The integer part has the following relation :


            , n is a natural number

            , n is a natural number



    Note that the meaning of "rounding operation" in computer programs is different from the meaning of "integer part" here : it was consistent at the time ; it was inconsistent at the time , for example , but after rounding on the computer .

    [ Divisibility ]   If there is an integer c , such that the integers a and b are suitable for

    Then b is said to be divisible by a , denoted as . In this case, a is called a multiple of b , and b is called a factor ( or submultiple ) of a.

If b is not divisible by a , it is recorded as b a .

    Divisibility has the following properties ( the following equations ):

    1 ° If , , then ; 

    2 ° If , then ; 

    3 ° If ​​, , then for any integer m,n we have 

    4 ° If b is a true factor ( ie ) of a, then 


    [ Prime numbers and Eratos sieve method ] An integer greater than 1 that has exactly 1 and itself two natural numbers as its factors is called a prime number , denoted as . Except for 2 , which is an even prime number , the other prime numbers are odd numbers .

    Prime numbers have the properties :

    There are infinitely many 1 ° prime numbers . If the number of prime numbers not exceeding the natural number n is denoted as p (n) , then at that time , there is * , and further there are 


    2 ° Let p be a prime number , if , then or . 

    The power of the square containing the prime number p in 3 ° is equal to 


    4 ° If it is a positive integer , it cannot be divisible by all prime numbers not exceeding , then n must be a prime number . This method of judging whether a natural number is a prime number is called the Eratosian sieve method . This method can establish a prime number table . 

    [ Unique Decomposition Theorem ] All natural numbers   greater than 1 can be uniquely decomposed into the product of prime powers . If , is a natural number , then n can be uniquely represented as


             ( for natural numbers )

               ( for prime numbers )

This is called the standard decomposition of n . The number s of different prime factors in n does not exceed .

    Obviously , any natural number n can be expressed as

                  ( k,m are natural numbers or zero )

This expression is unique .

    [ Mason number ]   integer

                  ( p is a prime number )

Those who are prime numbers are called Meissen numbers . So far only 27 have been found , namely


Whether there are infinite Meissen numbers has not yet been proved .

    [ Fermat number ]   integer

                    ( n is a natural number )

It is called the Fermat number . So far, only 5 Fermat numbers have been found to be prime , namely


None of the following 46 Fermat numbers are prime :


    [ Reversing and dividing method * ] Each integer a can be uniquely represented by a positive integer b as


In the formula, q is called the incomplete quotient obtained by dividing a by b , and r is called the remainder obtained by dividing a by b . The rolling division method refers to the following finite series of equations :

                               ( 1 )

    Example 1 set a = 525, b = 231, according to the formula (1) , the following formulas and formulas can be listed : 



           arithmetic grass _                      


    [ The greatest common factor and the least common multiple ]   Let a and b be integers . A positive integer that can divide both a and b is called the common factor of a and b , and the largest one is called the greatest common factor of a and b * , denoted as _

Especially at that time , it is said that a and b are mutually prime .

    Let a and b be positive integers . A positive integer that is divisible by both a and b is called the common multiple of a and b, and the smallest one is called the least common multiple of a and b * , which is written as


    Let n positive integers , and define their greatest common factor by induction as


Its least common multiple is


    The greatest common factor and the least common multiple have the following properties :

    1 ° There are integers x, y, such that x , y can be specifically obtained by the rolling division method . It is also obtained by a series of equations of the rolling division method , namely


    Example 2 obtains (525,231)=21 from Example 1. Because the formula from Example 1 has 


So we get x = 4, y = - 9 .

    2 ° must exist for any two integers x, y .

    3 ° If , , then .

    4 ° if then

       If , then

    5 ° If a and b are two positive integers , they are their prime factors , and the standard decomposition formulas are





    6 °

    If 7 ° is a co-prime positive integer , that is , then




* In number theory, the natural logarithm is usuallywritten as.

* Qin Jiushao, an ancient Chinese mathematician(also known as Euclid's algorithm)in "Nine Chapters of the Shushu"1247

* In foreign books and periodicals,the greatest common divisor of a and b is often written as gcd( a , b ),the least common multiple of a and b is often written as lcm ( a , b ).

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