Solve the Pythagorean Tripple 3^x+4^x=5^x

  • Thread starter Diffy
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In summary, the conversation is about the problem 3^x + 4^x = 5^x, which is a Pythagorean triple. The FLT theorem states that the only possible integer solution for x is 2. The conversation then explores the possibility of solving the equation without just knowing the answer is 2, such as using the natural log or integration. It is concluded that there is no general algebraic way to solve the equation for any non-integer values of x. The conversation also mentions the Fermat-Wiles theorem and the possibility of using L'hopital's rule, but it is acknowledged that these methods may only work in special cases or through approximations.
  • #1
Diffy
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The problem is 3^x + 4^x = 5^x. I recognize that this is a pythagorean triple but I am curious as to how you could solve this without just knowing that it is a pythagorean triple.
 
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  • #2
FLT states that the only way all of those solutions could be integers is if x=2.
 
  • #3
I am sure Fermats Theorem says what you say it says, however I am not sure that answer my question. I am trying to see if there is a way to solve the equation I have posted without just knowing the answer is 2.
 
  • #4
Solve what equation? 3^2+4^2 = 5^2 is obviously true. Are you looking for a general way to generate a three positive integers (a,b,c) such that a^2+b^2=c^2?
 
  • #5
I guess I am looking for an algebraic way to solve 3^x + 4^x = 5^x for x. (sorry if I used the term algebraic incorrectly). So what I mean is take the natural log of both sides, then do this etc. Or perhaps integrate both sides, then do something else etc.

Is it possible?
 
  • #6
Diffy said:
I guess I am looking for an algebraic way to solve 3^x + 4^x = 5^x for x. (sorry if I used the term algebraic incorrectly). So what I mean is take the natural log of both sides, then do this etc. Or perhaps integrate both sides, then do something else etc.

Is it possible?

No. It's called Fermat's last theorem.

Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation [tex]a^x + b^x = c^x[/tex] for any integer value of x greater than two.

So, assuming x is an integer, the only possible answers are 0, 1, and 2. Obviously, x = 2 works and the rest fail.
 
  • #7
Fermat-Wiles theorem. Give the guy credit who proved it along with the one who conjectured it.
 
  • #8
Ok. Fine. Say I didn't use 3,4,and 5 as a, b, and c. HOw do we know there are no non integer solutions.

I'm making this up off the top of my head, so say we had 4^x + 10^x = 13.4534543^x How would you solve for x?

0, 1 and 2 don't work obviously, if there were any other solutions for x, how would we solve it, if at all...
 
  • #9
That is possibly a very good question...

Could we take the log of it?Maybe. Would the answer then make sense?... Maybe. The question
is... do we need an integer solution of x?
 
  • #10
[tex]
\begin{array}{1}
f(x) = 4^x + 10^x - 13.4534543^x \\
f(1) > 0 \\
f(2) < 0 \\
[/tex]
So by the intermediate value theorem, there should exist an [tex]x_0 \in [1,2] [/tex]such that
[tex]
f (x_0) = 0
[/tex]
So there is a solution, I just don't think it can be solved by standard methods. These kind of things can usually only be solved in special cases or through approximations.
 
  • #11
Char. Limit said:
That is possibly a very good question...

Could we take the log of it?Maybe. Would the answer then make sense?... Maybe. The question
is... do we need an integer solution of x?

No I don't require integer solutions.

L'hopital -- Thanks, I think I am finally getting the answer I was looking for. I think starting with such a well know triple threw people off.
 

FAQ: Solve the Pythagorean Tripple 3^x+4^x=5^x

How do you solve the Pythagorean Triple 3^x+4^x=5^x?

To solve this equation, you can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the equation can be rewritten as (3^x)^2 + (4^x)^2 = (5^x)^2. This can be simplified to 9^x + 16^x = 25^x. From here, you can use algebraic techniques to solve for x.

Is there a specific method for solving this type of equation?

Yes, there are several methods for solving this type of equation, including using logarithms, graphing the equation, or using trial and error. It is important to first understand the Pythagorean Theorem and how it applies to this equation before attempting to solve it.

Can this equation have multiple solutions?

Yes, it is possible for this equation to have multiple solutions. Depending on the value of x, there may be more than one set of values that satisfy the equation. This is because there are infinitely many right triangles with sides that satisfy the Pythagorean Theorem.

Can this equation be solved for non-integer values of x?

Yes, this equation can be solved for non-integer values of x. While it may be more challenging to solve using algebraic techniques, you can use a scientific calculator or computer software to approximate the solution for non-integer values of x.

How is the Pythagorean Triple 3^x+4^x=5^x used in real life?

The Pythagorean Triple 3^x+4^x=5^x is used in various fields, such as engineering, architecture, and physics, to calculate the sides of a right triangle. It is also commonly used in construction to ensure that corners of buildings are square, as well as in navigation and surveying to measure distances and angles.

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