Pythagorean quadruple

Last updated October 16, 2024

A Pythagorean quadruple is a tuple of integers $a$ , $b$ , $c$ , and $d$ , such that $a 2 + b 2 + c 2 = d 2$ . 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.

Parametrization of primitive quadruples

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 ${\begin{aligned}a&=m^{2}+n^{2}-p^{2}-q^{2},\\b&=2(mq+np),\\c&=2(nq-mp),\\d&=m^{2}+n^{2}+p^{2}+q^{2},\end{aligned}}$ 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 $(m^{2}+n^{2}+p^{2}+q^{2})^{2}=(2mq+2np)^{2}+(2nq-2mp)^{2}+(m^{2}+n^{2}-p^{2}-q^{2})^{2}.$

Alternate parametrization

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 $a 2 + b 2$ such that $p 2 < a 2 + b 2$ . Then $c = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠a2 + b2 − p2/2p⁠$ and $d = ⁠ a 2 + b 2 + p 2 / 2 p ⁠$ . 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 $l 2 + m 2$ such that $n 2 < l 2 + m 2$ . Then $c = ⁠ l 2 + m 2 - n 2 / n ⁠$ and $d = ⁠ l 2 + m 2 + n 2 / n ⁠$ . 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.

Properties

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 $(a,b,c,d)$ where $d^{2}=a^{2}+b^{2}+c^{2}$ then $d$ can be defined as the norm of the quadruple in that $d={\sqrt {a^{2}+b^{2}+c^{2}}}$ 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 $d^{2}=a^{2}+b^{2}+c^{2}$ such that $a,b,c$ 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: ${\begin{aligned}x&=d^{2}-a^{2}\\y&=d^{2}-b^{2}\\z&=d^{2}-c^{2}\end{aligned}}$ 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: ${\begin{aligned}r_{x}&=bcd/a,\\r_{y}&=acd/b,\\r_{z}&=abd/c.\end{aligned}}$ The circumradius will be: $R={\frac {(d^{2}-a^{2})(d^{2}-b^{2})(d^{2}-c^{2})}{4abcd}}={\frac {abcd(1/a^{2}+1/b^{2}+1/c^{2}-1/d^{2})}{4}}$

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

Relationship with quaternions and rational orthogonal matrices

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

Primitive Pythagorean quadruples with small norm

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

(	1	,	2	,	2	,	3	)	(	2	,	10	,	11	,	15	)	(	4	,	13	,	16	,	21	)	(	2	,	10	,	25	,	27	)
(	2	,	3	,	6	,	7	)	(	1	,	12	,	12	,	17	)	(	8	,	11	,	16	,	21	)	(	2	,	14	,	23	,	27	)
(	1	,	4	,	8	,	9	)	(	8	,	9	,	12	,	17	)	(	3	,	6	,	22	,	23	)	(	7	,	14	,	22	,	27	)
(	4	,	4	,	7	,	9	)	(	1	,	6	,	18	,	19	)	(	3	,	14	,	18	,	23	)	(	10	,	10	,	23	,	27	)
(	2	,	6	,	9	,	11	)	(	6	,	6	,	17	,	19	)	(	6	,	13	,	18	,	23	)	(	3	,	16	,	24	,	29	)
(	6	,	6	,	7	,	11	)	(	6	,	10	,	15	,	19	)	(	9	,	12	,	20	,	25	)	(	11	,	12	,	24	,	29	)
(	3	,	4	,	12	,	13	)	(	4	,	5	,	20	,	21	)	(	12	,	15	,	16	,	25	)	(	12	,	16	,	21	,	29	)
(	2	,	5	,	14	,	15	)	(	4	,	8	,	19	,	21	)	(	2	,	7	,	26	,	27	)

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References

1 2 R. Spira, The diophantine equation $x 2 + y 2 + z 2 = m 2$ , Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.
↑ R. A. Beauregard and E. R. Suryanarayan, Pythagorean boxes, Math. Magazine 74 (2001), 222–227.
↑ R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.
↑ L.E. Dickson, Some relations between the theory of numbers and other branches of mathematics, in Villat (Henri), ed., Conférence générale, Comptes rendus du Congrès international des mathématiciens, Strasbourg, Toulouse, 1921, pp. 41–56; reprint Nendeln/Liechtenstein: Kraus Reprint Limited, 1967; Collected Works 2, pp. 579–594.
↑ Sierpiński, Wacław, Pythagorean Triangles , Dover, 2003 (orig. 1962), p.102–103.
↑ MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", Mathematical Gazette 96, March 2012, pp. 91-96.
↑ "OEIS A005818". The On-Line Encyclopedia of Integer Sequences.
↑ "OEIS A367737". The On-Line Encyclopedia of Integer Sequences.
↑ J. Cremona, Letter to the Editor, Amer. Math. Monthly 94 (1987), 757–758.

External links

Weisstein, Eric W. "Pythagorean Quadruple". MathWorld .
Weisstein, Eric W. "Lebesgue's Identity". MathWorld .
Carmichael. Diophantine Analysis at Project Gutenberg

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[Spira-1] 1 2 R. Spira, The diophantine equation $x 2 + y 2 + z 2 = m 2$ , Amer. Math. Monthly Vol. 69 (1962), No. 5, 360–365.

[2] R. A. Beauregard and E. R. Suryanarayan, Pythagorean boxes, Math. Magazine 74 (2001), 222–227.

[3] R.D. Carmichael, Diophantine Analysis, New York: John Wiley & Sons, 1915.

[4] L.E. Dickson, Some relations between the theory of numbers and other branches of mathematics, in Villat (Henri), ed., Conférence générale, Comptes rendus du Congrès international des mathématiciens, Strasbourg, Toulouse, 1921, pp. 41–56; reprint Nendeln/Liechtenstein: Kraus Reprint Limited, 1967; Collected Works 2, pp. 579–594.

[5] Sierpiński, Wacław, Pythagorean Triangles , Dover, 2003 (orig. 1962), p.102–103.

[6] MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", Mathematical Gazette 96, March 2012, pp. 91-96.

[7] "OEIS A005818". The On-Line Encyclopedia of Integer Sequences.

[8] "OEIS A367737". The On-Line Encyclopedia of Integer Sequences.

[9] J. Cremona, Letter to the Editor, Amer. Math. Monthly 94 (1987), 757–758.

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