**Learning Goal: **I’m working on a mathematics question and need an explanation and answer to help me learn.

4. In class we talked about the Four Color Theorem: if X is a planar graph with a finite number of vertices, then X can be colored using only 4 colors R, B, G and Y . In other words, each vertex of the graph can be assigned R, B, G or Y so that when two vertices are joined by an edge, those two vertices must have different colors. Take a look at the graphs on the next page.

(a) Find a coloring of the graph labeled A that uses 4 colors. Also, explain why no fewer colors will work.

(b) Explain why the graph labeled B cannot be colored with 4 colors. This graph is, however, not planar. Explain why it does not violate the Four Color Theorem. It might be helpful to think about the truth table of P =⇒ Q.

5. In class we talked about Fermat’s Last Theorem. In this problem, we show that x 2 + y 2 = z 2 has an infinite number of positive integer solutions (x, y, z) where (x, y, z) have no common factors. We already know about the solutions (3, 4, 5) and (5, 12, 13) and perhaps (8, 15, 17) since they show up all the time in high school. We call a solution (x, y, z) to the equation a primitive Pythagorean triple when (x, y, z) have no common factors.

(a) Let a and b be two positive integers with a > b. Show that setting x = a 2 − b 2 , y = 2ab, and z = a 2 + b 2 give a solution to the equation x 2 + y 2 = z 2 .

(b) What values of a and b yield the three Pythagorean triples above?

(c) Try your best to argue that if a and b have a common prime factor p, then x, y, and z will all have p as a factor.

(d) What common factor will x, y, z share if a and b have the same parity, meaning they are both even or both odd?

(e) In fact, if we avoid the two situations above, then (x, y, z) will be primitive. Use this result to find an infinite number of primitive Pythagorean triples.

(f) (Bonus) In fact, every primitive Pythagorean triple is obtained by this method. The first step in proving this is to see that for any primitive solution (x, y, z), exactly one of x and y must be even. In particular, x and y cannot both be odd. In class (and in the book in Example 1.21), we saw that if x is odd, then x 2 − 1 is divisible by 8, or equivalently, x 2 is of the form 8k + 1 for some integer k. Use this result (a theorem) to show that if x and y are both odd, then x 2 + y 2 is even, but never divisible by 4. On the other hand, if z 2 = x 2 + y 2 and x and y are both odd, explain why z 2 is divisible by 4. Complete the proof that if x 2 + y 2 = z 2 , then x and y cannot both be odd.

Why Work with Us

Top Quality and Well-Researched Papers

We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.

Professional and Experienced Academic Writers

We have a team of professional writers with experience in academic and business writing. Many are native speakers and able to perform any task for which you need help.

Free Unlimited Revisions

If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account or by contacting our support.

Prompt Delivery and 100% Money-Back-Guarantee

All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. In case you cannot provide us with more time, a 100% refund is guaranteed.

Original & Confidential

We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text. We also promise maximum confidentiality in all of our services.

24/7 Customer Support

Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.

Try it now!

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

No need to work on your paper at night. Sleep tight, we will cover your back. We offer all kinds of writing services.

Essays

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.

Admissions

Admission Essays & Business Writing Help

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.

Reviews

Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.

Reviews

Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.