# example of Fermat’s last theorem

Fermat stated that for any $n>2$ the Diophantine equation^{} ${x}^{n}+{y}^{n}={z}^{n}$ has no solution in positive integers. For $n=4$ this follows from the following

###### Theorem 1.

${x}^{4}+{y}^{4}={z}^{2}$ has no solution in positive integers.

###### Proof.

Suppose we had a positive $z$ such that ${x}^{4}+{y}^{4}={z}^{2}$ holds. We may assume $\mathrm{gcd}(x,y,z)=1$. Then $z$ must be odd, and $x,y$ have opposite parity. Since ${({x}^{2})}^{2}+{({y}^{2})}^{2}={z}^{2}$ is a primitive Pythagorean triple^{}, we have

$${x}^{2}=2pq,{y}^{2}={q}^{2}-{p}^{2},z={p}^{2}+{q}^{2}$$ | (1) |

where $p,q\in \mathbb{N}$, $$ are coprime^{} and have opposite parity. Since ${y}^{2}+{p}^{2}={q}^{2}$ is a primitive Pythagorean triple, we have coprime $s,r\in \mathbb{N}$, $$ of opposite parity satisfying

$$q={r}^{2}+{s}^{2},y={r}^{2}-{s}^{2},p=2rs.$$ | (2) |

From $\mathrm{gcd}({r}^{2},{s}^{2})=1$ it follows that $\mathrm{gcd}({r}^{2},{r}^{2}+{s}^{2})=1=\mathrm{gcd}({s}^{2},{r}^{2}+{s}^{2})$, which implies $\mathrm{gcd}(rs,{r}^{2}+{s}^{2})=1$. Since ${\left(\frac{x}{2}\right)}^{2}=\frac{pq}{2}=rs({r}^{2}+{s}^{2})$ is a square, each of $r,s,{r}^{2}+{s}^{2}$ is a square.

Setting ${Z}^{2}=q$, ${X}^{2}=r$, ${Y}^{2}=s$ $q={r}^{2}+{s}^{2}$ leads to

$${Z}^{2}={X}^{4}+{Y}^{4}$$ | (3) |

where $$. Thus, equation 3 gives a solution where $$. Applying the above steps repeatedly would produce an infinite^{} sequence^{} $z>Z>{z}_{2}>\mathrm{\dots}$ of positive integers, each of which was the sum of two fourth powers. But there cannot be infinitely many positive integers smaller than a given one; in particular this contradicts to the fact that there must exist a smallest $z$ for which (1) is solvable. So there are no solutions in positive integers for this equation.
∎

A consequence of the above theorem is that the area of a right triangle^{} with integer sides is not a square; equivalently, a right triangle with rational sides has an area which is not the square of a rational.

Title | example of Fermat’s last theorem |
---|---|

Canonical name | ExampleOfFermatsLastTheorem |

Date of creation | 2013-03-22 14:09:51 |

Last modified on | 2013-03-22 14:09:51 |

Owner | Thomas Heye (1234) |

Last modified by | Thomas Heye (1234) |

Numerical id | 9 |

Author | Thomas Heye (1234) |

Entry type | Example |

Classification | msc 11F80 |

Classification | msc 14H52 |

Classification | msc 11D41 |

Related topic | InfiniteDescent |

Related topic | X4Y4z2HasNoSolutionsInPositiveIntegers |