Math Is Fun Forum

Identity

Member

Registered: 2007-04-18

Posts: 934

Cool trick

This is a little trick I have found incredibly useful (especially in a past competition). The explanation is messy and bad, so please forgive me, I'm a noob at number theory too, so this is new for me. I'd be happy if someone could help clean up the explanation

If you have an equation of the form

With variables x,y and constants a,b,c,

It can be turned into the form:

Now introduce a new variable d such that

I find this very important as it allows you to be able to work out the possible numbers rather easily. The only real 'trick' which separates this from normal factorisation is the introduction of the new variable which is in ratio.

E.g Find all non-hypotenuse side lengths a,b of a right triangle, such that the hypotenuse of that triangle is equal to its area.

Now you can systematically work out the possible values of a and b. In this case they might not be rational but you get the idea.

Last edited by Identity (2007-08-30 18:19:56)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Cool trick

There is no need for the +/-, unless you are requiring b to be positive.  Also, I don't think it really allows you to work out the solutions.  For your example, I see no next step to take unless of course we are assuming that a and b must be integers.

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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..."

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Cool trick

Well, unless we assume either a or b are integers we will get an infinite number of solutions. If we assume one to be an integer, we can work upwards starting from the smallest values of a.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Cool trick

I think the middle part should look like this:

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

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Cool trick

It appears I've stumbled upon simon's favorite factoring trick:

http://www.artofproblemsolving.com/Wiki … ring_Trick

Last edited by Identity (2007-09-29 01:40:05)