Fermat's Last Theorem. The diophantine equation
$$x^{n}+y^{n}=z^{n}$$
has no solutions in nonzero integers $x, y, z$ when $n$ is an integer with $n \geqslant 3$.
Currently, we know that Fermat's last theorem is true for all positive integers $n$ with $3 \leqslant n \leqslant 125000$. In this section, we will show that the special case of Fermat's last theorem with $n=4$ is true. That is, we will show that the diophantine equation
$$x^{4}+y^{4}=z^{4}$$
has no solutions in nonzero integers $x, y, z$. Note that if we could also show that the diophantine equations
$$x^{p}+y^{p}=z^{p}$$
has no solutions in nonzero integers $x, y, z$ whenever $p$ is an odd prime, then we would know that Fermat's last theorem is true (see problem 2 at the end of this section).