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Prime Number

(1) Definition (Prime Number) A prime number is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

(2) Definition (Composite Number) A positive integer greater than 1 that is not prime is called a composite number.

(3) Example (Sieve of Eratosthenes) Using the sieve of Eratosthenes, find the prime numbers less than 100.

    Solution.    
    

(4) Proposition (Prime Divisor) Every positive integer greater than 1 has a prime divisor.

(5) Proposition (Infinitude of Primes) There are infinitely many primes.

    Proof. Assume that the prime numbers are listed in ascending order with and so on; and assume (for a contradiction) that is the last prime. Consider the integer,



Since every positive integer greater than 1 has a prime divisor, has a prime divisor, say However, can not divide since is of the form, Therefore, , the last prime number cannot exist.

(6) Proposition (Bounding Factors) If is a composite integer, then has a prime factor not exceeding

(7) Definition (Greatest Common Divisor) Let and be two integers that are not both zero. Then the greatest common divisor of and is the largest integer that divides both and The greatest common divisor of and is denoted by or sometimes by

(8) Example (Greatest Common Divisor) The integers 4 and 8 have greatest common divisor of 4 and we write since 4 is the largest integer that divides both 4 and 8. Also and

(9) Definition (Relatively Prime) If then and are said to be relatively prime.

(10) Example (Relatively Prime) Note that and so 2 and 5 are relatively prime. But and so 8 and 24 are not relatively prime.  

(11) Definition (Linear Combination) A linear combination of and is an integer of the form where and are integers.

(12) Example (Linear Combination) Note that given 2 and 51 we can form many different linear combinations, here are three examples: and

(13) Proposition (Properties of Greatest Common Divisors)  Let and be integers, then

    (i) for some integers and
    
    (ii) the set of linear combinations of and is the set of multiples of
    
    (iii) if and only if there exists integers and such that
    
    (iv) is the least positive integer that is a linear combination of and
    
    (v) if then
    
    (vi) if and only if and if is another common divisor then
    
    Proof. (i) Consider the set



Since is nonempty, ( are in ), the Well-Ordering Principle yields a least positive integer such that for some integers   and The idea is to show that To do this we use the Division Algorithm obtaining and such that where If then is in because   But we can not have is in S because and is the least in Therefore and so Using the same argument with replaced by it is shown that To show it remains to show that is greater than any other common divisor of and and so let be a common divisor of and Then using Properties of Divisibility, that is and so
     (ii) By part (i), if then for some integers and Thus every multiple of is also a linear combination of and Conversely, let be a linear combination of and Then using Properties of Divisibility, and so   is a multiple of
     (iii) If then by there exists integers and such that Conversely, suppose and that for some integers and Then because and so
     (iv) This follows from the Well Ordering Principle as in the proof of (i).
     (v) If then for some integers and Since and we have and by (ii)
     (vi) If then by definition, and Let be a common divisor of and with and for some integers and   By (i), there exists and such that and therefore we have   Whence,

(14) Example (Showing Relatively Prime) Show that and are relatively prime.

    Solution. Since it follows that

(15) Definition (Mutually Relatively Prime) The integers are called mutually relatively prime when

(16) Example (Mutually Relatively Prime) The integers 3, 5, 7 are mutually relatively prime and the integers 4, 19, 27 are mutually relatively prime.  

(17) Definition (Pairwise Relatively Prime) The integers are called pairwise relatively prime when for every possible and with

(18) Example (Pairwise Relatively Prime) The integers 3, 5, 7 are pairwise relatively prime because and The integers 4, 19, 27 are pairwise relatively prime because and
    The integers 10, 14, and 35 are mutually relatively prime because (there is no common divisor for all three) however they are not pairwise relatively prime because    

(19) Example (Great Common Divisors) What is where and are relatively prime integers that are not both 0?

    Solution. Let be a prime dividing Then



Now if then since But so Similarly, Therefore, and so or If and have the same parity, then and and so But if and have opposite parity, then and

(20) Example (Great Common Divisors) Show that if is an integer, then the integers and are relatively prime.

    Solution. Suppose that Then We have so if it follows that Hence To show that it is sufficient to show that neither 2 nor 3 divides If or and then and However, there are no such pairs in the set     

(21) Example (Great Common Divisors) Show that every positive integer greater than 6 is the sum of two relatively prime integers greater than 1.

    Solution. By the previous example, we know that   and are pairwise relatively prime. To represent as the sum of two relatively prime integers greater than one, let We now examine the twelve cases, one for each possible value of in the following table:
    

Number Theory Books

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