Euler's conjecture

The Euler's conjecture from 1769 is one of Leonhard Euler called presumption of number theory and generalizes the Fermat conjecture . Euler's conjecture has now been refuted, while the Fermatian conjecture has been proven.

guess

Euler's conjecture says that there are no positive integer solutions to the equation for . Fermat supposedly proved the conjecture for . Euler could find neither a proof nor a counterexample for larger ones. ${\displaystyle a_{1},a_{2},\dotsc ,a_{n}\in \mathbb {N} }$ ${\displaystyle a_{1}^{n}+a_{2}^{n}+\dotsb +a_{n-1}^{n}=a_{n}^{n}}$ ${\displaystyle n\geq 3}$ ${\displaystyle n=3}$ ${\displaystyle n}$

Refutations

Case n = 5

For the case , L. J. Lander and T. R. Parkin found a counterexample in 1966: ${\displaystyle n=5}$

${\displaystyle 27^{5}+84^{5}+110^{5}+133^{5}=144^{5}}$

Case n = 4

For 1988 Noam Elkies found the following counterexample: ${\displaystyle n=4}$

${\displaystyle 2.682.440^{4}+15.365.639^{4}+18.796.760^{4}=20.615.673^{4}}$

Elkies also proved that there are an infinite number of solutions. ${\displaystyle n=4}$

The smallest solution for is ${\displaystyle n=4}$

${\displaystyle 95.800^{4}+217.519^{4}+414.560^{4}=422.481^{4}}$.

This minimal solution was found after the publication of the first solution by Elkies from Roger Frye.

Related question

Together with his conjecture, Euler also stated that it should be possible to find four 4th powers, the sum of which results in a 4th power. This assumption was answered positively by R. Norrie in 1911:

${\displaystyle 30^{4}+120^{4}+272^{4}+315^{4}=353^{4}}$

For this general form

${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=e^{4}}$

was shown in 2008 by Lee W. Jacobi and Daniel J. Madden to have an infinite number of positive integer solutions. It also became a particularly aesthetic solution to the form

${\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}}$

found in whole numbers:

${\displaystyle 955+1770+(-2634)+5400=5491}$

${\displaystyle 955^{4}+1770^{4}+(-2634)^{4}+5400^{4}=5491^{4}}$

This equation is also called the Jacobi-Madden equation .

literature

• Richard K. Guy: Unsolved problems in number theory. Springer, New York 1994, ISBN 0-387-94289-0 .
• Ian Stewart, David Tall: Algebraic Number Theory and Fermat's Last Theorem . 3. Edition. AK Peters, Natick MA 2002, ISBN 1-56881-119-5 .

Web links

• Tito Piezas III: A Collection of Algebraic Identities .
• Computing Minimal Equal Sums Of Like Powers . EulerNet
• Jaroslaw Wroblewski: Equal Sums of Like Powers .
• Eric W. Weisstein : Euler's Sum of Powers Conjecture . In: MathWorld (English).
• Ivars Peterson: Euler's Sums of Powers. In: ScienceNews , 2004.

Individual evidence

1. ↑ LJ Lander, TR Parkin: Counterexample to Eulers's conjecture on sums of like powers. In: Bull. Amer. Math. Soc. Volume 72, 1966, p. 1079.
2. ↑ Noam Elkies: On . ${\displaystyle A^{4}+B^{4}+C^{4}=D^{4}}$In: Math. Comput. Volume 51, 1988, pp. 825-835.
3. ^ Ian Stewart, David Tall: Algebraic Number Theory and Fermat's Last Theorem . 3. Edition. AK Peters, Natick MA 2002, ISBN 1-56881-119-5 , pp. 232 .
4. ↑ Ivars Peterson: Euler's Sums of Powers . ( Memento from December 1, 2012 in the Internet Archive ) In: ScienceNews , 2004.
5. ^ American Mathematical Monthly. March 2008.
6. ↑ nzz.ch