MATH 319 Assignment #1 N² + 1

Due Monday, February 3.

This is an exploratory assignment. Your goal is to find patterns in numbers of a certain form. There are some obvious (obvious after you have stared at the data awhile) theorems that can be stated about these numbers.

1. For each positive integer n, starting with 1 and going up to 30 or so list n² + 1.

2. Which of these numbers are primes? Can you state a simple theorem about when numbers of the form n² + 1 are or are not primes?

3. Factor all the numbers you find. Are there any small primes that do not occur as factors in any of the numbers in your list?

4. Make two lists. The first should be all small primes that do occur as factors of n² + 1. The second is all small primes that do not. A small prime is one that is less than 100.

You can do all this by hand but either a TI-89 or Maple can be a big help.

TI-89

The function factor( in the algebra menu on the TI-89 will find the prime factorization of any integer. For large integers (40 digits or more) it might take longer than the battery life to do it. For integers of fewer than 10 digits it will factor almost instantaneously. It will factor expressions. For example if you want to factor 28² + 1 it will do that in that form. factor(28^2+1).

Maple

Maple is a little harder to use but not much. When you start Maple type in with(numtheory);. The semicolon is very important. This loads the number theory functions. Then just type ifactor(28^2+1); and it will give you the factorization of that number. divisors(28^2+1); will give a list of all the positive divisors of that number.