Pythagorean quadruple

A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a² + b² + c² = d². They are solutions of a Diophantine equation and often only positive integer values are considered.^[1] However, to provide a more complete geometric interpretation, the integer values can be allowed to be negative and zero (thus allowing Pythagorean triples to be included) with the only condition being that d > 0. In this setting, a Pythagorean quadruple (a, b, c, d) defines a cuboid with integer side lengths |a|, |b|, and |c|, whose space diagonal has integer length d; with this interpretation, Pythagorean quadruples are thus also called Pythagorean boxes.^[2] In this article we will assume, unless otherwise stated, that the values of a Pythagorean quadruple are all positive integers.

A Pythagorean quadruple is called primitive if the greatest common divisor of its entries is 1. Every Pythagorean quadruple is an integer multiple of a primitive quadruple. The set of primitive Pythagorean quadruples for which a is odd can be generated by the formulas $${\displaystyle {\begin{aligned}a&=m^{2}+n^{2}-p^{2}-q^{2},\\b&=2(mq+np),\\c&=2(nq-mp),\\d&=m^{2}+n^{2}+p^{2}+q^{2},\end{aligned}}}$$ where m, n, p, q are non-negative integers with greatest common divisor 1 such that m + n + p + q is odd.^[3][4][1] Thus, all primitive Pythagorean quadruples are characterized by the identity $${\displaystyle (m^{2}+n^{2}+p^{2}+q^{2})^{2}=(2mq+2np)^{2}+(2nq-2mp)^{2}+(m^{2}+n^{2}-p^{2}-q^{2})^{2}.}$$

All Pythagorean quadruples (including non-primitives, and with repetition, though a, b, and c do not appear in all possible orders) can be generated from two positive integers a and b as follows:

If a and b have different parity, let p be any factor of a² + b² such that p² < a² + b². Then c = ⁠a² + b² − p²/2p⁠ and d = ⁠a² + b² + p²/2p⁠. Note that p = d − c.

A similar method exists^[5] for generating all Pythagorean quadruples for which a and b are both even. Let l = ⁠a/2⁠ and m = ⁠b/2⁠ and let n be a factor of l² + m² such that n² < l² + m². Then c = ⁠l² + m² − n²/n⁠ and d = ⁠l² + m² + n²/n⁠. This method generates all Pythagorean quadruples exactly once each when l and m run through all pairs of natural numbers and n runs through all permissible values for each pair.

No such method exists if both a and b are odd, in which case no solutions exist as can be seen by the parametrization in the previous section.

The largest number that always divides the product abcd is 12.^[6] The quadruple with the minimal product is (1, 2, 2, 3).

Given a Pythagorean quadruple ${\displaystyle (a,b,c,d)}$ where ${\displaystyle d^{2}=a^{2}+b^{2}+c^{2}}$ then ${\displaystyle d}$ can be defined as the norm of the quadruple in that ${\displaystyle d={\sqrt {a^{2}+b^{2}+c^{2}}}}$ and is analogous to the hypotenuse of a Pythagorean triple.

Every odd positive number other than 1 and 5 can be the norm of a primitive Pythagorean quadruple ${\displaystyle d^{2}=a^{2}+b^{2}+c^{2}}$ such that ${\displaystyle a,b,c}$ are greater than zero and are coprime.^[7] All primitive Pythagorean quadruples with the odd numbers as norms up to 29 except 1 and 5 are given in the table below.

Similar to a Pythagorean triple which generates a distinct right triangle, a Pythagorean quadruple will generate a distinct Heronian triangle.^[8] If a, b, c, d is a Pythagorean quadruple with ${\textstyle a^{2}+b^{2}+c^{2}=d^{2}}$ it will generate a Heronian triangle with sides x, y, z as follows: $${\displaystyle {\begin{aligned}x&=d^{2}-a^{2}\\y&=d^{2}-b^{2}\\z&=d^{2}-c^{2}\end{aligned}}}$$ It will have a semiperimeter ${\textstyle s=d^{2}}$, an area ${\textstyle A=abcd}$ and an inradius ${\textstyle r=abc/d}$.

The exradii will be: $${\displaystyle {\begin{aligned}r_{x}&=bcd/a,\\r_{y}&=acd/b,\\r_{z}&=abd/c.\end{aligned}}}$$ The circumradius will be: $${\displaystyle R={\frac {(d^{2}-a^{2})(d^{2}-b^{2})(d^{2}-c^{2})}{4abcd}}={\frac {abcd(1/a^{2}+1/b^{2}+1/c^{2}-1/d^{2})}{4}}}$$

The ordered sequence of areas of this class of Heronian triangles can be found at (sequence A367737 in the OEIS).

A primitive Pythagorean quadruple (a, b, c, d) parametrized by (m, n, p, q) corresponds to the first column of the matrix representation E(α) of conjugation α(⋅)α by the Hurwitz quaternion α = m + ni + pj + qk restricted to the subspace of quaternions spanned by i, j, k, which is given by $${\displaystyle E(\alpha )={\begin{pmatrix}m^{2}+n^{2}-p^{2}-q^{2}&2np-2mq&2mp+2nq\\2mq+2np&m^{2}-n^{2}+p^{2}-q^{2}&2pq-2mn\\2nq-2mp&2mn+2pq&m^{2}-n^{2}-p^{2}+q^{2}\\\end{pmatrix}},}$$ where the columns are pairwise orthogonal and each has norm d. Furthermore, we have that ⁠1/d⁠E(α) belongs to the orthogonal group ${\displaystyle SO(3,\mathbb {Q} )}$, and, in fact, all 3 × 3 orthogonal matrices with rational coefficients arise in this manner.^[9]

There are 31 primitive Pythagorean quadruples in which all entries are less than 30.

• Beal conjecture
• Euler brick
• Euler's sum of powers conjecture
• Euler-Rodrigues formula for 3D rotations
• Fermat cubic
• Jacobi–Madden equation
• Lagrange's four-square theorem (every natural number can be represented as the sum of four integer squares)
• Legendre's three-square theorem (which natural numbers cannot be represented as the sum of three squares of integers)
• Prouhet–Tarry–Escott problem
• Quaternions and spatial rotation
• Taxicab number

• Weisstein, Eric W. "Pythagorean Quadruple". MathWorld.
• Weisstein, Eric W. "Lebesgue's Identity". MathWorld.

• Carmichael. Diophantine Analysis at Project Gutenberg