-
Recent Posts
Archives
- March 2023
- January 2023
- May 2020
- April 2020
- 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
Tag Archives: number theory
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 ≥ 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
Order of an Element Modulo m and Applications – Part II
Having introduced the concept of the (multiplicative) order of a modulo m, let us use it to solve some problems. Problem 1. Prove that if n > 1 is an integer, then n does not divide 2n – 1. Proof. … Continue reading
Order of an Element Modulo m and Applications – Part I
Background required : modular arithmetic If you’ve any experience observing powers of numbers, you’d have noticed that the last digit runs in cycles: e.g. if you take the last digits of successive powers of 7, you get 7 → 9 → … Continue reading
Homework (29 Oct 2011)
The homework for last week was a little harder than the prior one: Let n be a positive integer, . Prove that the sum of the divisors of n is a multiple of 24. Let N = 210 × 39 … Continue reading
Number Theory and Calculus/Analysis
Background required: modular arithmetic, calculus. Once in a while, I’ll post something which offers a glimpse into more advanced mathematics. Here’s one. Example 1 For starters, we know from basic algebra that . Let’s see if there’s a corresponding result … Continue reading
Number Theory Notes (22 Oct 2011) – Part III
Finally, we shall solve two more problems – the last problem is rather surprising since at first glance, it doesn’t appear to involve congruences. Problem 4 : Prove that if n is a perfect square, then . Solution : this is rather … Continue reading
Number Theory Notes (22 Oct 2011) – Part II
Now we will solve actual problems with the theory we’ve just learnt. Problem 1 : Find all integers x such that x ÷ 5 has remainder 3, x ÷ 7 has remainder 6 and x ÷ 9 has remainder 2. Solution : This can be … Continue reading
Number Theory Notes (22 Oct 2011) – Part I
Background required: none. For the first lecture, we shall look at congruence and modular arithmetic. Many of you may have already known (at least on an intuitive level) that square integers can only end in 0, 1, 4, 5, 6, … Continue reading
Mathematics and Such 