Number theory- prove no three ppt's with same value c

In summary, the problem is that the student is trying to find three different primitive pythagorean triples with the same value of c, but has not been able to prove it. He has been able to find a few triples, but is still looking for a more definitive solution.
  • #1
RossH
75
0

Homework Statement


The problem is that I have to prove that there aren't three or more primitive pythagorean triples with the same value of c. A primitive pythagorean triple has has no values, a, b, or c that have common factors.
The actual question is if this is possible, and if not prove it.

Homework Equations


Of course you know that a pythagorean triple fulfills the equation a2+b2=c2.
I am pretty sure that a relevant equation is the way to find pythagorean triples: a=st, b=(s2-t2)/2 c=(s2+t2)/2 for any s and to such that the above all are whole numbers.

The Attempt at a Solution


So far I have just been manipulating the various variables that I have above. I am trying to do a proof by contradiction, perhaps by creating a system of equations and showing that two of the triples must be identical, but all that I have managed to prove so far is that 0=0, which isn't exactly useful. I don't really know where to start if this isn't the right approach.

Thanks!
 
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  • #2
You can't prove that. Because it's not true. I'm not sure how you are expected to discover it's not true. Any idea?
 
  • #3
Dick said:
You can't prove that. Because it's not true. I'm not sure how you are expected to discover it's not true. Any idea?

Oops. Yeah, I just found a few triples. Thanks. I'm handing in the assignment tomorrow so I'll post back here what my professor says about that question.
 
  • #4
I use computer to compute from 1 to 1000, I still can't find 3 different ppts with same values of c. Can you give me an example?
 
  • #5
lifom said:
I use computer to compute from 1 to 1000, I still can't find 3 different ppts with same values of c. Can you give me an example?

I don't think you went quite far enough. Try 1105. And look at http://www.math.rutgers.edu/~erowland/pythagoreantriples.html
 
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FAQ: Number theory- prove no three ppt's with same value c

What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It is often referred to as the "Queen of Mathematics" because of its foundational role in other areas of mathematics.

What does it mean to prove that there are no three perfect power triples with the same value c?

This means that we are trying to show that there are no three distinct sets of three positive integers (p,q,r) that satisfy the equation c = p^a + q^b + r^c, where a, b, and c are all positive integers. In other words, we are trying to prove that there are no three perfect powers (numbers raised to a positive integer power) that add up to the same value.

Why is it important to prove that no three perfect power triples have the same value c?

This result has applications in several areas of mathematics, including algebraic number theory and diophantine equations. It also has implications in cryptography and coding theory.

What methods can be used to prove that no three perfect power triples have the same value c?

There are several approaches that can be used, including proof by contradiction, proof by infinite descent, and the use of modular arithmetic. Each method has its own advantages and may be more suitable for certain types of problems.

Can this result be extended to more than three perfect power triples having the same value c?

Yes, this result can be extended to any number of perfect power triples. It is known as Fermat's Last Theorem, which states that for any positive integer n greater than 2, the equation x^n + y^n = z^n has no positive integer solutions for x, y, and z. This was famously proven by Andrew Wiles in 1994.

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