Pythagorean Quadruple (original) (raw)

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A Pythagorean quadruple is a set of positive integers a, b, c, and d that 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).

Examples of primitive Pythagorean quadruples include (1,2,2,3), (2,3,6,7), (4,4,7,9), (1,4,8,9), (6,6,7,11), and (2,6,9,11).

Oliverio (1996) gives the following generalization of this result. Let S=(a_1,...,a_(n-2)), where a_i are integers, and let T be the number of odd integers in S. Then iff T≢2 (mod 4), there exist integers a_(n-1) and a_n such that

a_1^2+a_2^2+...+a_(n-1)^2=a_n^2. (2)

A set of Pythagorean quadruples is given by

where m,n, and p are integers (Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37).

See also

Diophantine Equation--4th Powers, Euler Brick, Pythagorean Triple, Sum of Squares Function

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References

Carmichael, R. D. Diophantine Analysis. New York: Wiley, 1915.Dutch, S. "Power Page: Pythagorean Quartets." https://stevedutch.net/recmath/rmpowers.htm#pythquart.Mordell, L. J. Diophantine Equations. London, England: Academic Press, 1969.Oliverio, P. "Self-Generating Pythagorean Quadruples and N-tuples." Fib. Quart. 34, 98-101, 1996.

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Pythagorean Quadruple

Cite this as:

Weisstein, Eric W. "Pythagorean Quadruple." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PythagoreanQuadruple.html

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