slides04 - Number Theory CMSC 250 1 Introductory number...

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1 CMSC 250 Number Theory
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2 CMSC 250 Introductory number theory A good proof should have: a statement of what is to be proven (labeled as “Theorem”, “Lemma”, “Proposition”, or “Corollary”) "Proof" to indicate where the proof starts a clear indication of flow a clear indication of the reason for each step careful notation, completeness and order a clear indication of the conclusion QED (or equivalent) to indicate end of proof
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3 CMSC 250 Some definitions Z is the integers Q is the rational numbers (quotients of integers) Irrational numbers are those which are not rational R is the real numbers A superscript of + indicates the positive portion only of one of these sets of numbers A superscript of indicates the negative portion only of one of these sets of numbers Other superscripts can be used, such as Z even , Z odd , Q >5 We can define the closure of these sets for an operation Z is closed under what operations?
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4 CMSC 250 Integer definitions An even integer n Z even ( 5 k Z ) [n = 2k] An odd integer n Z odd ( 5 k Z ) [n = 2k + 1] A prime integer (Z >1 ) n Z prime ( 2200 r,s Z + ) [ (n = r × s) (r = 1) (s = 1)] A composite integer (Z >1 ) n Z comp ( 5 r,s Z + ) [(n = r × s) (r 1) (s 1)]
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5 CMSC 250 Constructive proof of existence How can we prove the following? ] s
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slides04 - Number Theory CMSC 250 1 Introductory number...

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