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Tag Archives: notes
Modular Arithmetic Deluxe Edition
[ Background required: standard modular arithmetic. ] Consider the following two problems: Problem 1. Prove that if p > 2 is prime, then when is expressed in lowest terms , m must be a multiple of p. Problem 2. Prove that if … Continue reading
Posted in Notes
Tagged basic, congruence, modular arithmetic, notes, number theory, rational numbers
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Linear Algebra: Inner Products
[ Background required: basic knowledge of linear algebra, e.g. the previous post. Updated on 6 Dec 2011: added graphs in Application 2, courtesy of wolframalpha.] Those of you who already know inner products may roll your eyes at this point, … Continue reading
Posted in Notes
Tagged advanced, extra, fourier transform, inner product, intermediate, linear algebra, notes, sums
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Estimating Sums Via Integration
Background required : calculus, specifically integration By representing a sum as an area, it is often possible to estimate its size by approximating it with the area underneath a curve. For example, suppose we wish to compute the sum . … Continue reading
Matrices and Linear Algebra
Background recommended : coordinate geometry Here I thought I’d give an outline of linear algebra and matrices starting from a more axiomatic viewpoint, instead of merely giving rules of computation – the way it’s usually taught in school. The materials … Continue reading
Sample Problem Solving + Homework Hints
In this post, I’ll talk about basic number theory again. But I’ll still assume you already know modular arithmetic. 🙂 In the first part, there’ll be some sample solutions for number theoretic problems, some of which were already presented in … Continue reading
Quadratic Residues – Part IV (Applications)
Let p be an odd prime and g be a primitive root modulo p. Given any a which is not a multiple of p, we can write for some r. We mentioned at the end of the last section that a is a square if … Continue reading
Quadratic Residues – Part III
Ok, here’s the third installation. Getting a little tired of repeatedly saying “a is/isn’t a square mod p“, we introduce a new notation. Definition. Let p be an odd prime and a be an integer coprime to p. The Legendre symbol … Continue reading
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Tagged intermediate, legendre symbol, notes, number theory, quadratic residues
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Quadratic Residues – Part II
Recall what we’re trying to do here: to show that if p > 2 is prime and a is not a square modulo p, then . As mentioned at the end of the previous part, we will need… Primitive Roots Again, let … Continue reading