30 Introduction to Number Theory

30 Introduction to Number Theory - Handout #30 Feb. 13,...

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Handout #30 CS103A Feb. 13, 2008 Robert Plummer Introduction to Number Theory Basics of Number Theory Number theory was once thought to be “pure” mathematics – math for the sake of math. But with the evolution of computers (and particularly cryptography), number theory has become known as an applied area of math. Much cryptology relies on some very old theorems from number theory. What is number theory? It is the branch of mathematics that studies problems about natural numbers, integers and rational numbers, i.e., anything but real numbers. It was invented by the Greeks. Number theory is a subject in which the concepts are simple but the problems can be quite challenging. We choose to work with number theory for two reasons in 103A: a) The simple axioms and definitions in number theory allow us to focus on proof techniques and not get distracted; b) Number theory has many important applications in CS. We assume for our purposes that you understand what integers are: negative and positive whole numbers including 0. Note that 0 is considered to be neither positive nor negative. We also assume that you understand and accept as valid the basic arithmetic operations. The slides from today's lecture discuss these points in more detail. The integers are closed under addition, subtraction and multiplication, meaning if we add/subtract/multiply an integer and another integer, we get an integer. The integers, however, are not closed under division, which sets division apart from other operations. We can still consider “div” an integer operation if we define it in a special way. (Note: where proofs are not given and not covered in lecture, you are encouraged to find your own.) Definition : If a and b are integers and a 0, then the statement that a divides b means that there is an integer c such that b = ac. Our notation for “a divides b” is a | b. Theorem 30.1 : For all integers a, b, and d, if d | a and d | b then d | (a ± b) Theorem 30.2: For all integers a, b, and c, if a | b then a | bc. Proof: Suppose a | b, then there exists an integer k such that ak = b. Since (ak)c = bc, we can conclude that a | bc, since there exists an integer (kc) such that a(kc) = bc. Q.E.D. Theorem 30.3: For all integers a, b, and c, if a | b and b | c then a | c.
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2 The divisibility definition leads us to the Division Theorem, which is an elegant term for something you have known since elementary school. We can perform division working only with integers if we introduce the notion of a remainder: The Division Theorem Let a be an integer and let d be a positive integer. Then there exist unique integers q and r, with 0 r < d such that a = dq + r. r is called the remainder of a divided by d and q is called the quotient . This theorem is often called the Division Algorithm. One of the oldest (yet still useful) applications of the Division Theorem is in finding greatest
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This note was uploaded on 10/01/2011 for the course CS 103A taught by Professor Plummer,r during the Winter '07 term at Stanford.

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30 Introduction to Number Theory - Handout #30 Feb. 13,...

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