# Generalized Diophantine equation and the method of infinite descent

• A
• e2m2a
In summary, it has been proven that there are no integer solutions to the Diophantine equation ## x^{2p} + y^{2p} = z^2 ## for all values of ## p \geq 2 ##. The proof for this can be found in the cited paper. It is not clear if the method of infinite descent was used in this proof.
e2m2a
TL;DR Summary
Cannot find proof asserted by Wikipedia article on a generalized Diophantine equation
There is an entry in Wikipedia at this link: https://en.wikipedia.org/wiki/Pythagorean_triple
Under elementary properties of primitive Pythagorean triples, general properties,sixth bullet from the bottom of this section, there is this generalized Diophantine equation:
x^2p + y^2p = z^2
Where: p ≥ 2.
The article asserts there is no integer solution to this Diophantine equation for all values of p ≥ 2:
I have a number of questions about this. First, is this assertion true? Second, where can I find the proof for this? There was no citation for a proof. And third, If there is a proof, would it use the method of infinite descent for this generalized expression?

e2m2a said:
sixth bullet from the bottom
• Only two sides of a primitive Pythagorean triple can be simultaneously prime because by Euclid's formula for generating a primitive Pythagorean triple, one of the legs must be composite and even.[20] However, only one side can be an integer of perfect power ##{\displaystyle p\geq 2}## because if two sides were integers of perfect powers with equal exponent ##{\displaystyle p}## it would contradict the fact that there are no integer solutions to the Diophantine equation ##{\displaystyle x^{2p}\pm y^{2p}=z^{2}}##, with ##{\displaystyle x}, {\displaystyle y}, ## and ##{\displaystyle z}## being pairwise coprime.[21]
Make life easier for all by quoting in stead of referring ... I't not a long entry entry in the list
(and some understand first from bottom is one but last )

e2m2a said:
First, is this assertion true?
I don't understand what assertion you are referring to. The Beal conjecture lemma clearly states that it hasn't been proved or disproved so far. So no wonder there is no citation. And the 'can be simultaneously prime' is on top of that conjecture.

The link says you can earn a million bucks if you can prove or disprove the Beal conjecture

e2m2a said:
Second, where can I find the proof for this? There was no citation for a proof

##\ ##

pbuk
e2m2a said:
There was no citation for a proof.
Yes there is: click on the blue number 21. The cited paper is available online (search for the title).

Ok. I will look into it. Thanks. Beal conjecture sounds intriguing.

BvU said:
Make life easier for all by quoting in stead of referring ... I't not a long entry entry in the list
(and some understand first from bottom is one but last )
Thanks for this @BvU - as well as being easier, quoting also avoids the problem of the referenced website changing. @e2m2a please quote instead of linking in future. Also please use ## \LaTeX ## in your posts: write ## x^{2p} + y^{2p} = z^2 ## instead of x^2p + y^2p = z^2 (if you don't know how, reply to this message and you will see how it works in my quoted message).

BvU said:
I don't understand what assertion you are referring to. The Beal conjecture lemma clearly states that it hasn't been proved or disproved so far. So no wonder there is no citation. And the 'can be simultaneously prime' is on top of that conjecture.

The link says you can earn a million bucks if you can prove or disprove the Beal conjecture
I think the OP is referring to the special case of the Beal conjecture quoted above which has been solved (H. Darmon and L. Merel (2007) Winding quotients and some variants of Fermat’s Last Theorem).

BvU

## 1. What is a Generalized Diophantine equation?

A Generalized Diophantine equation is a mathematical equation in which the unknown variables are required to be integers. These equations often involve multiple variables and exponents, and can be difficult to solve using traditional algebraic methods.

## 2. What is the method of infinite descent?

The method of infinite descent is a mathematical proof technique used to show that a Diophantine equation has no solutions. It involves starting with a hypothetical solution and then repeatedly reducing it until it reaches a contradiction. If a contradiction is reached, it proves that the original equation has no solutions.

## 3. How is the method of infinite descent used to solve Diophantine equations?

The method of infinite descent is not used to solve Diophantine equations, but rather to prove that they have no solutions. This is useful in situations where finding a solution is not possible or practical, such as in Fermat's Last Theorem.

## 4. What are some real-world applications of Generalized Diophantine equations?

Generalized Diophantine equations have various applications in fields such as cryptography, number theory, and computer science. They are also used in the study of prime numbers and in finding solutions to certain types of equations in physics and engineering.

## 5. Are there any limitations to the method of infinite descent?

The method of infinite descent can only be used to prove that a Diophantine equation has no solutions, not to find actual solutions. It also requires a high level of mathematical understanding and can be difficult to apply in certain situations. Additionally, it cannot be used to prove the existence of solutions, only their non-existence.

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