mod5 - Module 5: Basic Number Theory Theme 1: Division...

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Module 5: Basic Number Theory Theme 1: Division Given two integers, say and ,thequo t ien t may or may not be an integer (e.g., but ). Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers. More formally, for we say that divides if there is another integer such that and we write .Inshor t : if and only if This simple deFnition leads to many properties of divisibility. ±or example, let us establish the following lemma. Lemma 1 If and ,then . Proof .W eg i v ead i r e c tp roo f romth ed eFn i t iono fd i v i s ib i l
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Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number is an integer greater than that is divisible only by and itself. A number that is not prime is called composite . Example 1 :Thepr imeslessthan are: How many primes are there? We Frst prove that there are inFnite number of primes. Theorem 1 . There are infnite number oF primes. Proof .W ep r o v i d eap r o o fb yc o n t r a d i c t i o n .A c t u a l l y ,i ti sd u et oE u c l i da n di
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where are exponents of (i.e., the number of times occurs in the factorization of ) . Proof
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This note was uploaded on 03/26/2012 for the course STAT 350 taught by Professor Staff during the Spring '08 term at Purdue University.

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mod5 - Module 5: Basic Number Theory Theme 1: Division...

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