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Pythagorean Quadruple

Positive Integers $a$, $b$, $c$, and $d$ which satisfy

\end{displaymath} (1)

For Positive Even $a$ and $b$, there exist such Integers $c$ and $d$; for Positive Odd $a$ and $b$, no such Integers exist (Oliverio 1996). Oliverio (1996) gives the following generalization of this result. Let $S=(a_1, \ldots, a_{n-2})$, where $a_i$ are Integers, and let $T$ be the number of Odd Integers in $S$. Then Iff $T\not\equiv 2$ (mod 4), there exist Integers $a_{n-1}$ and $a_n$ such that
\end{displaymath} (2)

A set of Pythagorean quadruples is given by

$\displaystyle a$ $\textstyle =$ $\displaystyle 2mp$ (3)
$\displaystyle b$ $\textstyle =$ $\displaystyle 2np$ (4)
$\displaystyle c$ $\textstyle =$ $\displaystyle p^2-(m^2+n^2)$ (5)
$\displaystyle d$ $\textstyle =$ $\displaystyle p^2+(m^2+n^2),$ (6)

where $m$, $n$, and $p$ are Integers,
m+n+p\equiv 1\ \left({{\rm mod\ } {2}}\right),
\end{displaymath} (7)

\end{displaymath} (8)

(Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37). Another set of solutions can be obtained from
$\displaystyle a$ $\textstyle =$ $\displaystyle 2mp+2nq$ (9)
$\displaystyle b$ $\textstyle =$ $\displaystyle 2np-2mq$ (10)
$\displaystyle c$ $\textstyle =$ $\displaystyle p^2+q^2-(m^2+n^2)$ (11)
$\displaystyle d$ $\textstyle =$ $\displaystyle p^2+q^2+(m^2+n^2)$ (12)

(Carmichael 1915).

See also Euler Brick, Pythagorean Triple


Carmichael, R. D. Diophantine Analysis. New York: Wiley, 1915.

Mordell, L. J. Diophantine Equations. London: Academic Press, 1969.

Oliverio, P. ``Self-Generating Pythagorean Quadruples and $N$-tuples.'' Fib. Quart. 34, 98-101, 1996.

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© 1996-9 Eric W. Weisstein