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We know the following facts:

(1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$.

(2) For all $3\leq n$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has no solution in $\mathbb{N}$.

Question: Is the following generalization true?

For all $2\leq m$ both of the following statements are true:

(1) For all $1\leq n\leq m$ the equation $x_{1}^{n}+...+x_{m}^{n}=x_{m+1}^{n}$ has a solution in $\mathbb{N}$.

(2) For all $m+1\leq n$ the equation $x_{1}^{n}+...+x_{m}^{n}=x_{m+1}^{n}$ has no solution in $\mathbb{N}$.

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  • $\begingroup$ That has been first conjectured by Euler, but much later proved to be wrong, see the answers. $\endgroup$ – Michael Oct 26 '13 at 3:32
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No this is not true. For $n=4$ one has $$2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$$ found by Elkies (as part of an infinite family of solutions). Also earlier it was known, for example, for $n=5$, $$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$ by Lander and Parkin.

But part 2 was a conjecture of Euler so you are in good company, and $n \ge 6$ is still open. See that page for further details.

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    $\begingroup$ wow! The world of numbers is the world of wonders! Thanks dear quid. $\endgroup$ – user36136 Oct 25 '13 at 23:51
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    $\begingroup$ You are welcome! One thing to note is that direct extrapolation from Fermat can be risky since for $n=3$ there actually should be solutions on certain heuristic grounds. See an answer of Elkies to a question on heuristically false theorems for this and also information directly related to your question. $\endgroup$ – user9072 Oct 25 '13 at 23:57
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    $\begingroup$ Thanks for your useful references. I am just a passenger from Cantor's heaven in your wonderful realm. $\endgroup$ – user36136 Oct 26 '13 at 0:04
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The smallest counterexample for $n = 4$ is $95800^4 + 217519^4 + 414560^4 = 422481^4$. This has been found out by Roger Frye in 1988, cf. http://euler.free.fr/docs/euler88.ps.

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