Math Is Fun Forum

shaoen01

Member

Registered: 2008-01-26

Posts: 18

Number Theory

Hi all,

I am having some difficulty understanding the question below and did not quite get one part of the solution. It would be great if someone could guide me.

Qns:
Find the truth set of "

is prime" where the domain is N (special symbol N).

What i did and don't understand:
I know for a number m to be a prime number, m>1. I know that

. How exactly do i find out the truth set? What i don't understand about the solution is that it states n>2 is not prime. Why? How do i deduce that other than trying to substitute values in myself?

Thanks

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Number Theory

Well, you basically solve it in your first step right there.  Can a number be prime if it has a proper factor?  If you plug n into 3n^2 - 4n + 1, what two factors must that number have (based solely upon the work you already did)?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

shaoen01

Member

Registered: 2008-01-26

Posts: 18

Re: Number Theory

Hi Ricky,

Do pardon me if i didn't manage to catch what you were saying correctly. I know a number cannot be prime if it has a factor like numbers 4, 6 have factor like 2 and 3. And prime numbers can only be divided by themselves. Is that what you are trying to say? I know if i input values like n=3 or n=5 into the equation and it would not be prime answer. So i guess this is how it is proven?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Number Theory

You factorised the expression yourself, to (3n-1)(n-1).
The only time that that will be a prime is when either 3n-1 or n-1 is equal to 1.
Otherwise it would have two proper factors and not be prime, by definition.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Number Theory

The only time that that will be a prime is when either 3n-1 or n-1 is equal to 1.

Or −1. In some cases, it can be −1. For example, the problem might be when the following is a prime for n a natural number:

−n[sup]2[/sup]+10n−9 = (n−9)(1−n). n−9 = 1 and 1−n = 1 won’t work; you must try −1 instead.
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shaoen01

Member

Registered: 2008-01-26

Posts: 18

Re: Number Theory

Hi,

But how do you know that it will only be a prime when "3n-1 or n-1 is equal to 1"? Is something to do with the definition of prime, where m>1 and m=(r)(s), where r=1 or s=1?