Mths 640
HW #9
July 29, 2008
In case it was not entirely clear what the homework is for Tuesday, here is a
restatement of the problem.
You are asked to give an example of a map on a torus, which consist of seven
countries each of which shares a border wit
Mths 640
Solutions to HW #2
July 1, 2008
Problem: Prove that the Fermat numbers are given by the formula Fn =
n
22 :
Solution: The Fermat numbers F0 ; F1 ; F2 ; : were dened by dening
F0 = 3; F1 = 5;
and in general, for each number n > 0; dening
Fn = (F0
Mths 640
Solutions to HW #4
July 8, 2008
Problem 1: Consider the statement: Among all rectangular boxes
of given total edge length, the one with the largest volume is the
cube. Find an equivalent algebraic statement.
Solution: Suppose the three edge lengt
Mths 640
Hints for HW #2
Spring, 2008
Since the current homework may require technique you don (yet) possess,
t
I give a few suggestions here. In fact, both of the problems will yield to a
ll
method of proof called mathematical induction which will be of
Maths 640
Solutions to HW #1
June 26, 2008
Problem : Show that there are innitely many primes among the numbers
of the form 4k + 3; where k is a positive integer.
Before giving the solution, we note that the set of all whole numbers is
divided into four c
Mths 640
Solution to HW #5
July 10, 2008
Problem: Prove that there are no positive integers n and m such that
n2
= 3:
m2
Solution 1: Suppose that there were such positive integers n and m: Then
there would be a smallest positive integer m0 so that, for so
Mths 640
Solutions to HW #7
July 17, 2008
p
Problem: Show that the numbers 3; and i 6 are prime in the set consisting
p
of all numbers of the form m + in 6; where m and n are integers.
Solution: Let M denote this set of numbers. Recall that to say that a
Mths 640
Summer 2008
Here are two proofs of the fact that there are innitely many primes. Each of these proofs have at their
center an idea like the one that drives Euclid proof, which we discussed in class. Each of these proofs
s
can be dressed up to loo
Mths 640
The Sieve
Spring 2008
We can describe the prime numbers as those numbers that survive a certain
process of elimination. We begin with the denition:
Denition 1 A prime number is a number p (that is, a positive, whole number
p) which is strictly gr
Mths 640
Solutions to HW #8
July 22, 2008
p
Problem: Let M denote the set of numbers of the form m + in 6; where
m and n are integers. Show that every number in M (other than 0; 1; and 1)
is either prime or a product of primes.
Solution: Remember, a numbe
Mths 640
Solutions to HW #6
July 15, 2008
Problem 1: Inscribe a regular pentagon ABCDE in a circle, (the vertices
are labeled in order going around the circle), and prove that the segments AC
and CD are incommensurable. What number is thereby shown to be