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Homederivation of Pythagorean triples

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# derivation of Pythagorean triples

For finding all positive solutions of the Diophantine equation

$\displaystyle x^{2}\!+\!y^{2}\;=\;z^{2}$ | (1) |

we first can determine such triples $x,\,y,\,z$ which are coprime. When these are then multiplied by all positive integers, one obtains all positive solutions.

Let $(x,\,y,\,z)$ be a solution of the mentioned kind. Then the numbers are pairwise coprime, since by (1), a common divisor of two of them is also a common divisor of the third. Especially, $x$ and $y$ cannot both be even. Neither can they both be odd, since because the square of any odd number is $\equiv 1\;\;(\mathop{{\rm mod}}4)$, the equation (1) would imply an impossible congruence $2\equiv z^{2}\;\;(\mathop{{\rm mod}}4)$. Accordingly, one of the numbers, e.g. $x$, is even and the other, $y$, odd.

Write (1) to the form

$\displaystyle x^{2}\;=\;(z\!+\!y)(z\!-\!y).$ | (2) |

Now, both factors on the right hand side are even, whence one may denote

$\displaystyle z\!+\!y\;=:\;2u,\quad z\!-\!y\;=:\;2v$ | (3) |

giving

$\displaystyle z\;=\;u\!+\!v,\quad y\;=\;u\!-\!v,$ | (4) |

and thus (2) reads

$\displaystyle x^{2}\;=\;4uv.$ | (5) |

Because $z$ and $y$ are coprime and $z>y>0$, one can infer from (4) and (3) that also $u$ and $v$ must be coprime and $u>v>0$. Therefore, it follows from (5) that

$u\;=\;m^{2},\quad v\;=\;n^{2}$ |

where $m$ and $n$ are coprime and $m>n>0$. Thus, (5) and (4) yield

$\displaystyle x\;=\;2mn,\quad y\;=\;m^{2}\!-\!n^{2},\quad z\;=\;m^{2}\!+\!n^{2}.$ | (6) |

Here, one of $m$ and $n$ is odd and the other even, since $y$ is odd.

By substituting the expressions (6) to the equation (1), one sees that it is satisfied by arbitrary values of $m$ and
$n$. If $m$ and $n$ have all the properties stated above, then $x,\,y,\,z$ are positive integers and, as one may deduce from two first of the equations (6), the numbers $x$ and $y$ and thus all three numbers are coprime.

Thus one has proved the

Theorem. All coprime positive solutions $x,\,y,\,z$, and only them, are gotten when one substitutes for $m$ and $n$ to the formulae (6) all possible coprime value pairs, from which always one is odd and the other even and $m>n$.

# References

- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

## Mathematics Subject Classification

11D09*no label found*11A05

*no label found*

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