Lesson 4

Problem. Fermat's Last Theorem (formulated by Pierre de Fermat in 1637 and proved in 1994 by Andrew Wiles) states that the equation \[a^{n}+b^{n}=c^{n}\] has no solutions in positive integers for any integer \(n \ge 3\).

In 1769, Leonhard Euler proposed a hypothesis generalizing Fermat's theorem and stating that for \(n \in \mathbb{Z}_{>2}\), representing an \(n\)-th power as a sum of \(n\)-th powers requires at least \(n\) terms. In particular, the equations \[a^{4}+b^{4}+c^{4}=d^{4} \text{ and }a^{5}+b^{5}+c^{5}+d^{5}=e^{5}\] have no solutions for \(a,b,c,d,e \in \mathbb{Z}_{>0}\).

It turns out that Euler's hypothesis is false: there exists a counterexample. To disprove the hypothesis, enter four or five numbers (separated by spaces) that satisfy one of the equations above.