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January 3rd, 2011, 01:44 PM   #1
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Please help me understand the syntax

I got this problem I have to find the solution for:

Quote:
Find the no of distinct triplets(a,b,c) which satisfy the equation

a^n+b^n=c^n

a,b,c <=100 2<=n<=20 a,b,c are non-negative integers
Please note that I am NOT asking for the answer of this problem.

I just don't fully understand the question, I'll just note down what I think it is I have to do, just correct me if i'm wrong.

My theory: I have to find every unique combination of variables A, B and C. (the count of all these combinations is the answer)
A, B and C can have values ranging from 0~100.
N can be any value ranging from 2~20

Is this correct? do I have the ranges correct? or am I missing something completely?
Please help me out with this problem, thank you
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January 3rd, 2011, 02:08 PM   #2
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Re: Please help me understand the syntax

The problem assumes you are naive about mathematics. Fermat's last theorem (proved around 1995) states that for n > 2 there are no solutions. So the problem is really finding the solutions for n=2.

The general approach is to let i and j be any relatively prime integers with i > j. Then a=i^2 - j^2, b=2ij will give you a solution. As you can see, c=i^2 + j^2.

http://en.wikipedia.org/wiki/Formulas_f ... an_triples

The above is a more complete discussion.
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January 3rd, 2011, 02:34 PM   #3
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Re: Please help me understand the syntax

Yes, I'm pretty naive about this subject but I'm trying.

So you're saying there are no other values for n that would make a match besides the value 2?
also I tried figuring out the relatively prime solution but I can't seem to figure out how they work.

I realize I'm being a pain because my lack of knowledge in this area, but I'd be very grateful if someone could explain me on what to do exactly.
I'ts a programming problem and I need to solve it using my programming abilities, but it's useless if I don't know what to do.

Please help me out.
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January 3rd, 2011, 04:32 PM   #4
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Re: Please help me understand the syntax

If it were me, I would approach this problem using the method of Euclid for generating Pythagorean triples. As [color=#008000]mathman[/color] stated, Fermat's Last Theorem was proven to be true, so only n = 2 need be considered. Let m and n be non-negative integers where and k be a positive integer:





and





So, for (m,n)=(0,0) we get the triple (0,0,0) for any value of k from both sets.

For (m,n) = (1,0) we get k(1,0,1) and k(0,1,1) so you can let k vary from 1 to 100.

For (m,n) = (1,1) we get k(0,2,2) and k(2,0,2) but these will not be distinct as the previous case has already generated them. Thus we no longer need check when m = n or when n = 0.

For (m,n) = (2,1) we get k(3,4,5) and k(4,3,5) so you can let k vary from 1 to 20.

For (m,n) = (3,1) we get k(8,6,10) and k(6,8,10) but these will already have been generated by the previous case. In fact, from here on out, we need only consider when m and n are co-prime and only one of them is odd.

Hope this helps you get started.
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January 4th, 2011, 10:22 AM   #5
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Re: Please help me understand the syntax

Your interpretation of the problem is correct.

Quote:
Originally Posted by pieter888
Yes, I'm pretty naive about this subject but I'm trying.
It's the problem that is naive, not you! It happens that there can be no solutions unless n = 2, but the problem is sneaky and doesn't mention this.
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