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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. ðŸ™‚Â 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 â€“ Part IV (Applications)
LetÂ pÂ be an odd prime and gÂ be a primitive root modulo p. Given any aÂ which is not a multiple of p, we can write for some r. We mentioned at the end of the last section that aÂ is a square if … Continue reading
Quadratic Residues â€“ Part III
Ok, here’s the third installation. Getting a little tired of repeatedly saying “aÂ 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
Posted in Notes
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 pÂ > 2 is prime and aÂ 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