Conjecture of Euler

In Mathematical, the conjecture of Euler , is a refuted Conjecture, but which was originally proposed by the Swiss mathematician Leonhard Euler in 1769, and which is stated in the following way:

For all whole N strictly higher than 2, the sum of N -1 powers N E is not a power N E.

In other words, and in a more formal way:

$\backslash \; forall\; N\; >\; 2,\; \backslash \; forall\; (a\_1,\; \backslash \; dowries,\; a\_\; \{n-1\})\; \backslash \; in\; (\backslash \; mathbb\; \{NR\}\; ^*)\; ^\; \{n-1\},\; \backslash \; forall\; m\; >\; 1,\; \backslash \; sum\_\; \{k=1\}\; ^\; \{n-1\}\; \{a\_k\}\; ^n\; \backslash \; m^n$

This conjecture was cancelled by L.J. Lander and T.R. Parkin in 1966 thanks to the following counterexample:

$27^5\; +\; 84^5\; +\; 110^5\; +\; 133^5\; =\; 144^5$.

In 1988, Noam Elkies found a method to build counterexamples when N = 4. Its simpler counterexample was the following:

$2682440^4\; +\; 15365639^4\; +\; 18796760^4\; =\; 20615673^4$.

Thereafter, Roger Frye found the smallest counterexample possible for N = 4 while using, with a Ordinateur, techniques suggested by Elkies:

$95800^4\; +\; 217519^4\; +\; 414560^4\; =\; 422481^4$.

No counterexample for N > 5 is currently known.

External bonds

• EulerNet: Minimal Computing Equal Sums Off Like Powers
• the conjecture quartic of Euler in MathWorld
• Equations diophantiennes of the fourth degree in MathWorld
• the conjecture of Euler about the site library.thinkquest.org
• a simple explanation of the conjecture of Euler about the site Maths Is Good For You!

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