# Monthly Archives: November 2011

## Thoughts on a Problem II

The following problem caught my eye: (USAMO 1997 Q3) Prove that for any integer n, there is a unique polynomial Q(X) whose coefficients all lie in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and Q(-2) … Continue reading

## 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

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## 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

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## 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

## 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

## Quadratic Residues – Part I

[ Background required : modular arithmetic. Seriously. ] Warning: many of the proofs for theorems will be omitted in this set of notes, due to the length of the proofs. The basic question we’re trying to answer in this series … Continue reading