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1. people.math.ethz.ch › ~kowalski › number-theoryNumber Theory I - people.math.ethz.ch

   people.math.ethz.ch › ~kowalski › number-theory

   CHAPTER 1 Introduction 1.1. What is number theory? 1.2. Looking at sums of two squares We will start by considering sums of two squares of (positive) integers. This is an extremely classical subject, but there are still many research questions connected to it, some of which will be mentioned below. However, we begin with a natural beginning for

2. artofproblemsolving.com › articles › filesNumber Theory - Art of Problem Solving

   artofproblemsolving.com › articles › files

   Example 1.1. Let x and y be integers. Prove that 2x + 3y is divisible by 17 iff 9x+5y is divisible by 17. Solution. 17 | (2x + 3y) ⇒ 17 | [13(2x + 3y)], or 17 | (26x + 39y) ⇒ 17 | (9x + 5y), and conversely, 17 | (9x + 5y) ⇒ 17 | [4(9x + 5y)], or 17 | (36x+20y) ⇒ 17 | (2x+3y). Example 1.2. Find all positive integers d such that d ...

3. people.math.harvard.edu › ~knill › teachingLecture 4: Number Theory - Harvard University

   people.math.harvard.edu › ~knill › teaching

   Number theory studies the structure of prime numbers and equations involving integers. Gauss called it the \Queen of Mathematics". We look here at a few theorems as well as some open problems in this eld. Figure 1.

4. www.math.brown.edu › ~jhs › MA0076Introduction to Higher Mathematics Unit #4: Number Theory

   www.math.brown.edu › ~jhs › MA0076

   Number Shapes. The square numbers are the numbers 1, 4, 9, 16, ... that can be arranged in the shape of a square. The triangular numbers are the num-bers 1, 3, 6, 10, ... that can be arranged in the shape of a triangle. The first few triangular and square numbers are illustrated in Figure 1.1. 1+2 = 3 1+2+3 = 6 1+2+3+4 = 10

5. www.math.u-bordeaux.fr › ~hecohen › numthrybookNumber Theory - u-bordeaux.fr

   www.math.u-bordeaux.fr › ~hecohen › numthrybook

   In addition to the purely analytical problems, the theory of L-functions contains beautiful results (and conjectures) on special values, of which Euler’s formula Pn>1 1/n2 = π2/6 is a special case. This book can be considered as having four main parts.

6. math.stanford.edu › ~conrad › 154PageMATH 154. ALGEBRAIC NUMBER THEORY - Stanford University

   math.stanford.edu › ~conrad › 154Page

   MATH 154. ALGEBRAIC NUMBER THEORY 7 and 4 = (1 + p 3)(1 p 3). We claim: Lemma 1.16. The element 2 is irreducible in Z[p 3]. Proof. Say 2 = ab, so by conjugating both sides we have 2 = ab. Multiply-ing both relations gives 4 = (aa)(bb). with aa and bb each a non-negative integer since for a = u + v p 3 with u,v 2Z we have aa = u2 +3v2. But u2 ...

7. www.claymath.org › library › proceedingsAnalytic Number Theory - Clay Mathematics Institute

   www.claymath.org › library › proceedings

   international gathering of leading number theorists who reported on recent advances in both classical analytic number theory as well as in related parts of number theory and algebraic geometry.