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

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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

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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

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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

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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

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