# 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  $\equiv 1\pmod{4}$, the equation (1) would imply an impossible congruence$2\equiv z^{2}\pmod{4}$.  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 (http://planetmath.org/Product) 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)

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

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).
Title derivation of Pythagorean triples DerivationOfPythagoreanTriples 2013-03-22 18:34:40 2013-03-22 18:34:40 pahio (2872) pahio (2872) 8 pahio (2872) Derivation msc 11D09 msc 11A05 LinearFormulasForPythagoreanTriples ContraharmonicMeansAndPythagoreanHypotenuses