A Pythagorean quadruple is a set of positive integers , , , and that satisfy
(1)
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For positive even and , there exist such integers and ; for positive odd and , no such integers exist (Oliverio 1996).
Examples of primitive Pythagorean quadruples include , , , , , and .
Oliverio (1996) gives the following generalization of this result. Let , where are integers, and let be the number of odd integers in . Then iff (mod 4), there exist integers and such that
(2)
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A set of Pythagorean quadruples is given by
(3)
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(4)
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(5)
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(6)
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where , , and are integers (Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37).