
Recent Posts
Archives
 March 2020
 June 2018
 July 2016
 June 2016
 May 2016
 March 2015
 February 2015
 January 2015
 December 2014
 December 2013
 November 2013
 July 2013
 June 2013
 May 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 April 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011
Categories
Meta
Pages
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
Leave a comment
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