Math 58 - Number Theory

September 12, 2006

Homework 3 (due September 14):

1. Find all primes of the form n³ + 1, where n is a positive integer.

2. Show that 200 is the sum of two primes.

3. Show that 220 is the sum of two primes.

4. Define binomial coefficients: If n ≥ 0 and 0 ≤ k ≤ n, then .

Show that if p is prime and 1 ≤ k ≤ p–1, then p | .

5. Here are six evenly spaced primes: 7, 37, 67, 97, 127, 157.

Can you find seven evenly spaced primes?

(This may be hard. Start by reading and, if you like, doing text exercises

15-19, page 87, and 10, page 89.)

Homework 4 (due September 19):

Prove that every integer greater than 1 can be written as a product of primes, and that except for the order of the primes, this representation is unique.

Use the axioms; but, you may assume anything you know to be true about addition, subtraction, multiplication, and ordering.

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