Gaussian Integers and their Relationship to Ordinary Integers Iris Yang and Victoria Zhang Brookline High School and Phillips Academy Mentor Matthew Weiss May 1920th 2018 MIT Primes Conference GOAL prove unique factorization for Gaussian integers and make comparisons to ordinary integers ID: 762000
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Gaussian Integers and their Relationship to Ordinary Integers Iris Yang and Victoria Zhang Brookline High School and Phillips Academy Mentor Matthew Weiss May 19-20th, 2018 MIT Primes Conference
GOAL: prove unique factorization for Gaussian integers (and make comparisons to ordinary integers) I ntroductionGaussian Prime TheoremEuclidean Algorithm; Greatest Common DivisorsGaussian Prime Divisibility PropertyUnique Prime Factorization Outline
Introduction
Gaussian Primes
Gaussian Prime Theorem: Proof
Gaussian Prime Theorem: Proof of (i) and (iii)
Gaussian Prime Theorem: Proof of (ii) contradiction
Gaussian Prime Theorem: Proof
Gaussian Prime Theorem: Gaussian Divisibility Lemma (a)
Quadratic Reciprocity
Quadratic Reciprocity: Definitions
Quadratic Reciprocity: Theorems
Gaussian Prime Theorem Gaussian Divisibility Lemma: Part (b)
Gaussian Prime Theorem: Gaussian Divisibility Lemma (b)
Gaussian Prime Theorem: Gaussian Divisibility Lemma (c) Show
Gaussian Prime Theorem: Gaussian Divisibility Lemma (c)
Gaussian Prime Theorem: Gaussian Divisibility Lemma (c)
Gaussian Prime Theorem: Gaussian Divisibility Lemma (c)
Gaussian Prime Theorem , a Gaussian Prime
Gaussian Prime Theorem
Euclidean Algorithm for Gaussian Integers show
Euclidean Algorithm for Gaussian Integers
Greatest Common Divisor
Greatest Common Divisor
Greatest Common Divisor Theorem: Proof
Greatest Common Divisor Theorem: Proof
Gaussian Prime Divisibility Property
Unique Factorization
Unique Factorization
Acknowledgements We would like to extend our great thanks to the following people:Matthew Weiss, our mentor The MIT PRIMES-Circle Program and Isabel Vogt Our teachers Our parents
Questions?
Thank You!