Fermat Last Theorem? Maybe NOT...

Prove that there are infinitely many positive integer solutions to \[x^n+y^n=z^{n+1}.\]

Solution: Let $p,q$ be natural numbers and $p^n+q^n=k$ , then we have $(p^n+q^n)^n(p^n+q^n)=k^{n+1}$ $(p(p^n+q^n))^n+(q(p^n+q^n))^n=k^{n+1}.\square$

Are there any other solutions that you can think of?

#NumberTheory

Easy Math Editor

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Comments

Is it necessary for $p\neq q \neq n$ . If it is, then plugging numbers $p=1,q=2,n=3$ gives $3^{3}+6^{3}=3^{5}$ and $p=n$ . Hence I think you must state some more conditions for it to work always.

Vinayak Srivastava - 1 year ago

@Vinayak Srivastava I made a mistake, I had edited it.

ChengYiin Ong - 10 months ago

Ohk, but the solution is nice, I am not able to think of any other method.

Vinayak Srivastava - 10 months ago

Interesting... @ChengYiin Ong

A Former Brilliant Member - 1 year ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$