Showing a Prime exists

Feb 2017
91
0
N/A
Professor decided to give more "optional" homework.

Let A = {4k +1 | k Z^+ }. Show that there exists a prime p of A, and elements j, k, of A, such that p | (jk), but p does not divide j or k.

In words: "Let A equal the set of elements of 4k+1, where k is an element of all positive integers. Show that there exists a prime p of A, and elements j, k of A, such that p divides j times k, but p does not divide j OR k"

I am aware that this equation (4k+1) does not have a Unique Factorization Property, but I am not sure if that has anything to do with the problem.
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey azollner95.

If p | jk then it means np = jk for some integer n.

Try building on those definitions to show that you will only have one of them to be prime and the other while have to be a 1.
 
Feb 2017
91
0
N/A
I understand that np = jk for some integer n. However, I am still unclear on how that helps me show that there exists a prime p of A.

Here is what I have so far:

"A does not have a Unique Factorization Property. Because of this, there is more than one set of primes of A. A is a subset of all positive integers and thus is closed under multiplication. x is an element of A if and only if j and k are elements of A, and x =jk, which implies j=x or k=x. Let j,k be elements of A, which are elements of all positive integers, with j not equal to zero. Then j divides k. Let p be a prime of A. Suppose p | (jk). Then either p | a or p | b."

Any suggestions/hints/tips/help on moving forward to prove that there exists a prime of A?
 
Last edited:

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
You have to think about what happens when something does not divide another thing.

In mathematics it's a good idea to list all the information you have [usually in terms of formulas] and then go from there.

I am trying to help you to list these points of information so you can go from taking a word problem and making it a series of mathematical statements for which you find solutions to.
 
Feb 2017
91
0
N/A
I'm sorry, but I am still beyond confused with this problem. I have left off with "Then either p | j or p | k. (not p|a and p|b as I miswrote earlier).
 
Feb 2017
91
0
N/A
I ended up figuring it out!

Solution: "Let p be a prime of A. Suppose p | (jk). Then either p | j or p | k. Let p ∈ Z^+. Let j=21 and k=33 and p=9. 9 does not divide 21 and 9 does not divide k, but 9 does divide 21 times 33. Thus there exists a prime p of A."