MHB Why does it suffice to show it for n=4 and n=p?

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The discussion centers on the Fermat equation \(X^n + Y^n = Z^n\) for \(n \geq 3\) and the assertion that proving it for \(n=4\) and \(n=p\) (where \(p\) is a prime not equal to 2) suffices to show there are no integer solutions. Participants explore the implications of the fundamental theorem of arithmetic and the concept of infinite descent in relation to the case of \(n=4\). They clarify that while all integers greater than 4 can be expressed as products of primes, the specific case of \(n=4\) requires different reasoning. The conversation highlights the complexity of proving the absence of solutions for specific values of \(n\) and invites further exploration of the methods involved.
evinda
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Hi! (Wave)

The Fermat equation $X^n+Y^n=Z^n, n \geq 3, X \cdot Y \cdot Z \neq 0$ has no integer solutions.

Remark: It suffices to prove the hypothesis for $n=4$ and $n=p \in \mathbb{P} (p \neq 2)$.

Could you explain me why, in order to show that the Fermat equation has no integer solutions, it suffices to show it for $n=4$ and for $n=p$ ? (Thinking)
 
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evinda said:
Hi! (Wave)

The Fermat equation $X^n+Y^n=Z^n, n \geq 3, X \cdot Y \cdot Z \neq 0$ has no integer solutions.

Remark: It suffices to prove the hypothesis for $n=4$ and $n=p \in \mathbb{P} (p \neq 2)$.

Could you explain me why, in order to show that the Fermat equation has no integer solutions, it suffices to show it for $n=4$ and for $n=p$ ? (Thinking)

Hint: if $x^{pq} + y^{pq} = z^{pq}$, then $(x^p)^q + (y^p)^q = (z^p)^q$. Use the fundamental theorem of arithmetic, suitably modified to take into account the $n = 4$ case, and conclude.
 
Just a note to why Bacterius's reasoning doesn't work for $n = 4$ :

$$X^4 + Y^4 = Z^4 \Longrightarrow (X^2)^2 + (Y^2)^2 = (Z^2)^2$$

But that doesn't conclude anything, as there can be infinitely many solutions in general to $x^2 + y^2 = z^2$.
 
Bacterius said:
Hint: if $x^{pq} + y^{pq} = z^{pq}$, then $(x^p)^q + (y^p)^q = (z^p)^q$. Use the fundamental theorem of arithmetic, suitably modified to take into account the $n = 4$ case, and conclude.

So, because of the fact that all numbers $n \geq 3, n \neq 4$ can be written as a product of prime numbers, at least one of which is different from $2$, we can show that $x^n+y^n=z^n$ has no integer solutions, when $n$ is a prime? (Thinking)

How can I use the fundamental theorem of arithmetic, to show that $X^4+Y^4=Z^4$ has no integer solutions? (Thinking)
 
evinda said:
So, because of the fact that all numbers $n \geq 3, n \neq 4$ can be written as a product of prime numbers, at least one of which is different from $2$, we can show that $x^n+y^n=z^n$ has no integer solutions, when $n$ is a prime? (Thinking)

Minor correction : it's because of all numbers $> 4$ can be written as a product of primes with at least one prime $> 2$.

How can I use the fundamental theorem of arithmetic, to show that $X^4+Y^4=Z^4$ has no integer solutions?

As a matter of fact you cannot, as I have indicated in my last comment. You need to use some of the "magic tricks" like infinite descent to handle this one. If you are interested, I could post the idea behind.
 
mathbalarka said:
Minor correction : it's because of all numbers $> 4$ can be written as a product of primes with at least one prime $> 2$.

(Nod)
mathbalarka said:
As a matter of fact you cannot, as I have indicated in my last comment. You need to use some of the "magic tricks" like infinite descent to handle this one. If you are interested, I could post the idea behind.

Yes, it would be nice.. (Smile)
 
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