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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
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
Number Theory Homework (2 Weeks)
Homework problems for 5 Nov 2011: Let a1, a2, … be a series recursively defined as follows: a1 = 20, a2 = 11, and for n â‰¥ 1, an+2 is the remainder when an+1 + an is divided by 100. … Continue reading
Thoughts on a Problem
From time to time, I may post my experience in solving a specific problem including some wrong twists and turns I’ve taken, as well as the motivation for some rather cryptic constructions. Warning : since this post documents my thought … Continue reading