Pythagorean Quadruple

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

$$a^2+b^2+c^2=d^2.$$ (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

$${a_1}^2+{a_2}^2+\ldots+{a_{n-1}}^2={a_n}^2.$$ (2)

A set of Pythagorean quadruples is given by

$$a = 2mp$$ (3)

$$b = 2np$$ (4)

$$c = p^2-(m^2+n^2)$$ (5)

$$d = p^2+(m^2+n^2),$$ (6)

where $m$, $n$, and $p$ are Integers,

$$m+n+p\equiv 1\ \left({{\rm mod\ } {2}}\right),$$ (7)

and

$$(m,n,p)=1$$ (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

$$a = 2mp+2nq$$ (9)

$$b = 2np-2mq$$ (10)

$$c = p^2+q^2-(m^2+n^2)$$ (11)

$$d = p^2+q^2+(m^2+n^2)$$ (12)

(Carmichael 1915).

See also Euler Brick, Pythagorean Triple

References

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.