Lecture 3 - Number Theory - Number Theory (Computer Science...

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Number Theory (Computer Science Notes) Number Theory Number theory is a branch of mathematics concerned with the behavior of integers. Factors and Multiple Definition: Suppose that a and b are integers. Then a divides b if b = an for some integer n. a is called a factor or divisor of b. b is called a multiple of a. The shorthand for a divides b is a | b. Be careful about the order. The divisor is on the left and the multiple is on the right (−3) | 12 because 12 = 3 · −4 7 | 7 because 7 = 7 · 1 A number p is even exactly when 2 | p. The fact that zero is even is just a special case of the fact that zero is divisible by any integer Direct Proof with Divisibility Claim 1 For any integers a,b,and c, if a | b and a | c then a | (b + c). Proof: Let a,b,and c and suppose that a | b and a | c. Since a | b, there is an integer k such that b = ak (definition of divides). Similarly, since a | c, there is an integer j such that c = aj. Adding these two equations, we find that (b + c) = ak + aj = a(k + j). Since k and j are integers, so is k + j. Therefore, by the definition of divides, a | (b + c) Claim 2 For any integers a, b, and c, if a | b and b | c then a | c. (Transitivity of divides.) Stay in the Set Students are sometimes tempted to rephrase the definition of a | b as “b a is an integer.” This is not a good idea because it introduces a non-integer rational number into a problem that only needs to involve integers. Stepping outside the set of interest in this way is occasionally useful, but more often it leads to inelegant and/or buggy solutions. When constructing math from the ground up, the integers are typically constructed first and the rationals built from them. So using rationals to prove facts about integers can lead to circular proofs On computers, integer operations yield exact answers but floating point operations are only approximate. So implementing an integer calculation using real numbers often introduces errors. Prime Numbers
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Lecture 3 - Number Theory - Number Theory (Computer Science...

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