Assignment, Week 1

Reading in Silverman

Introduction
Chapter 1: What is Number Theory?
Chapter 2: Pythagorean Triples

Problems

Solutions    Comments

Remember: A problems are required. The ones on the B list are optional. But if you want to get an A grade, you need to do some B problems over the course of the semester. There is no fixed due date for the B problems. Be sure to write these up in your notebook as well as submitting them.

A Problems

1.1., 1.2, 1.3 --- problems, 1, 2, 3, of chapter 1.

1.1: Square-triangular numbers.

Intro. A square number is one like 1, 4, 9, 16. Think of a counting a square array of dots. A triangular number is a number like 1, 3, 6, 10, etc. (Do you see the pattern? What is the next triangular number?). Triangular numbers come from counting triangular arrays of dots. (Make a little drawing in your notebook). Now here is the question. Which numbers are both square and triangular? Well, certainly 1 is. You'll find that 36 is too. Are there any smaller triangular numbers that are also square?

The problem from Silverman: Find the next square-triangular number after 36, and, if possible, the next one after that. Can you figure out an efficient way to find square-triangular numbers? Do you think that there are infinitely many?

1.2: Sums of odd numbers

The problem from Silverman: Try adding up the first few odd numbers, e.g, 1, 1 + 3, 1 + 3 + 5, and see if the numbers you get satisfy some sort of pattern. Once you find the pattern, express it as a formula. Give a geometric verification that the formula is correct.

1.3: Twin primes and prime triples

Intro. Primes are numbers like 2, 3, 5, 7, 11, 13, 17, 19, ... that have no divisors other than 1 and themselves. (Composites are numbers like 4, 6, 8, 9, etc. that have interesting divisors). As we shall see later (and as the Greeks knew), there are infinitely many primes. As you can see from this short list, and as you can see if you continue the list, primes often come in pairs. Examples are (3,5), (5,7), (11,13), (17,19). It is thought (conjectured!) that there are infintely many "twin primes.". However, to this day, no one has found a proof.

The problem from Silverman. The consecutive odd numbers 3, 5, 7 are all primes. Are there infinitely many "prime triples"? That is, are there infinitely many prime numbers p so that p+2 and p+4 are also primes?

B Problems

1. How many primitive Pythagorean triples (a,b,c) are there with a <= b <= c <= 10? Try to answer the same question with 10 replaced first by 20, then by 40. Describe how you obtained your answer. How would one solve this problem for 10 replaced by a fairly large integer N? What kind of difficulties does this problem raise?

The B problem is an open-ended question. I'm interested in seeing what you come up with.

1	1	1
2	4	3
3	9	6
4	16	10
5	25	15
6	36	21
7	49	28
8	64	36
9	81	45
10	100	55
111	121	66
12	144	78
13	169	91
...	...	...