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Pythagorean quadruple

From Wikipedia, the free encyclopedia

All four primitive Pythagorean quadruples with only single-digit values

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.

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Transcription

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

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

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 a2 + b2 such that p2 < a2 + b2. Then c = a2 + b2p2/2p and d = a2 + b2 + p2/2p. Note that p = dc (this method is based on results obtained on Pythagorean triples [5]).

A same 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 + m2n2/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.

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 where then can be defined as the norm of the quadruple in that 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 such that 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 it will generate a Heronian triangle with sides x, y, z as follows:-

.

It will have a semiperimeter , an area and an inradius .

The exradii will be:-

.

The circumradius will be

.

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(⋅)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

where the columns are pairwise orthogonal and each has norm d. Furthermore, we have that 1/dE(α) belongs to the orthogonal group , 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 )

See also

References

  1. ^ a b R. Spira, The diophantine equation x2 + y2 + z2 = m2, 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. ^ a b Amato, Roberto, A characterization of Pythagorean triples, JP Journal of Algebra, Number Theory and Applications 39 (2) (2017), 221-230.
  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.

External links

This page was last edited on 28 March 2024, at 09:29
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