Fermat's Theorm extension

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In summary, if n is a prime greater than 2 and x, y, z are prime to each other, then 2z^n(z^n + x^n + y^n) can never be a perfect square. However, if there is a perfect square of the form 2x^p(x^p+y^p+z^p) for p an odd prime, then 2x(x^p+y^p+z^p) is also a perfect square. Considering the case of cubes, it is possible for 2z^p(z^p + x^p + y^p) to be a perfect square even if x^p + y^p = z
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ramsey2879
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Is it true that [tex]2z^n(z^n + x^n + y^n)[/tex] can never be a perfect square if n is a prime greater than 2 and x,y,z are prime to each other?
 
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If there is a perfect square of the form [itex]2x^p(x^p+y^p+z^p)[/itex] for p an odd prime, then [itex]2x(x^p+y^p+z^p)[/itex] is also a perfect square, so consider that instead.

Since x, y, and z are coprime, at most one can be odd. If all three were odd then [itex]2x(x^p+y^p+z^p)\equiv2\pmod4[/itex] and so it is not a perfect square. Thus exactly one of x, y, and z is even -- and without loss of generality, we can assume that z is odd.
 
  • #3
ramsey2879: Is it true that [tex]2z^n(z^n + x^n + y^n)[/tex]
can never be a perfect square if n is a prime greater than 2 and x,y,z are prime to each other?


If x^p + y^p = z^p, then [tex]2z^p(z^p + x^p + y^p)[/tex]
=[tex]2z^p(2z^p)[/tex]

However, does this imply the converse? Not at all, consider this case of cubes:
x=2, y=3, z=5:

[tex](2)(5^3)(2^3+3^3+5^3)=250x 160 = 25(100)(16)=(200)^2[/tex]

Or look at: [tex](2)(19^3)(5^3+14^3+19^3) =(2)(19^4)(151+19^2)=(2^{10})(19^4)[/tex]
 
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What is "Fermat's Theorem extension"?

"Fermat's Theorem extension" refers to a generalization of Fermat's Last Theorem, which states that there are no positive integer solutions to the equation x^n + y^n = z^n for any integer value of n greater than 2. The "extension" refers to the exploration of solutions for non-integer values of n.

Why is "Fermat's Theorem extension" important?

"Fermat's Theorem extension" is important because it expands our understanding of Fermat's Last Theorem and offers potential solutions for equations that were previously unsolvable. It also has applications in various fields of mathematics, such as number theory and algebraic geometry.

What are some examples of "Fermat's Theorem extension"?

One example of "Fermat's Theorem extension" is the generalization of the theorem to equations with rational exponents. Another example is the exploration of solutions for equations with more than three variables, as opposed to the original formula with three variables.

What are the challenges in studying "Fermat's Theorem extension"?

One of the main challenges in studying "Fermat's Theorem extension" is the complexity of the equations involved. As n becomes a non-integer value, the equations become more difficult to solve and require advanced mathematical techniques. Another challenge is the lack of a unified approach to studying these extensions, as different methods may be required for different types of equations.

What are some potential applications of "Fermat's Theorem extension"?

The potential applications of "Fermat's Theorem extension" include the development of new mathematical techniques and tools, as well as the potential for solving previously unsolvable mathematical problems. It also has implications in fields such as cryptography and computer science.

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