Movable singularity - Revision history

en>Monkbot: Task 2: Fix CS1 deprecated coauthor parameter errors

2014-07-18T11:39:24Z

Task 2: Fix CS1 deprecated coauthor parameter errors

← Older revision		Revision as of 13:39, 18 July 2014
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	~~A '''[[Pythagoras\|Pythagorean]] quadruple'''~~ is ~~a [[tuple]] of [[integer]]s ''a'', ''b'', ''c''~~ and ~~''d'', such~~ that '~~'d > 0'' and <math>a^2 + b^2 + c^2 = d^2</math>, and~~ is ~~often denoted <math>(a,b,c,d)</math>. Geometrically, a Pythagorean quadruple <math>(a,b,c,d)</math> defines a [[cuboid]] with side [[Norm (mathematics)#Euclidean norm\|lengths]] \|''~~a~~''\|, \|''b''\|, and \|''c''\|, whose [[space diagonal]] has integer length ''d''~~. ~~Pythagorean quadruples are thus also called ''Pythagorean Boxes''.<ref>R.A. Beauregard and E. R. Suryanarayan, ''Pythagorean boxes'', Math. Magazine '''74''' (2001), 222–227.</ref>~~		Nice to satisfy you, my name is Ling and I totally dig that title. I presently live in Arizona but now I'm contemplating other options. My occupation is a messenger. One of my favorite hobbies is tenting and now I'm trying to earn cash with it.<br><br>Also visit my web-site :: [http://Www.Planetplayschool.com/ActivityFeed/MyProfile/tabid/61/UserId/415/Default.aspx planetplayschool.com]

	~~== Parametrization of primitive quadruples ==~~
	~~The [[Set (mathematics)\|set]] of all '''primitive''' Pythagorean quadruples, i.e., those for which gcd(''a'',''b'',''c'') = 1, where gcd denotes the [[greatest common divisor]]~~ of ~~''a'', ''b'', and ''c'',~~ is ~~parametrized by,<ref>R.D. Carmichael, ''Diophantine Analysis'', New York: John Wiley & Sons, 1915.</ref><ref>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.</ref><ref>R. Spira, ''The diophantine equation <math>x^2 + y^2 + z^2 = m~~^2</math>'', Amer. Math. Monthly '''69''' (1962), 360–365~~.<~~/ref~~>

	:<~~math~~> ~~a = m^2+n^2~~-~~p^2-q^2,\,</math>~~

	~~:<math> b = 2(mq+np),\,</math>~~

	:~~<math> c = 2(nq-mp),\,</math>~~

	:~~<math> d = m^2+n^2+p^2+q^2,\,</math>~~

	~~where ''m'', ''n'', ''p'', ''q'' are non-negative~~ [~~[integer]]s and gcd(''m'', ''n'', ''p'', ''q'') = 1 and ''m'' + ''n'' + ''p'' + ''q'' ≡ 1 (mod 2). Thus, all primitive Pythagorean quadruples are characterized by the [[Lebesgue Identity]]~~

	~~:<math>(m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2.</math>~~

	~~==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 <math>a</math> and <math>b<~~/~~math> have different parity, let ''p'' be any factor of <math>a^2 + b^2<~~/~~math> such that <math>p^2 < a^2 + b^2</math>~~. ~~Then <math>c = (a^2 + b^2 - p^2)/(2p)</math> and <math>d = (a^2 + b^2 + p^2)/(2p)</math>. Note that <math>p = {d - c}</math>.~~

	A similar method exists<ref>[[Wacław Sierpiński\|Sierpiński, Wacław]], ''Pythagorean Triangles'', Dover, 2003 (orig. 1962), p.102.</ref> for <math>a, b</math> both even, with the further restriction that <math>2p</math> must be an even factor of <math>a^2 + b^2</math>. No such method exists if both ''a'' and ''b'' are odd.

	~~==Properties==~~
	The biggest number that always divides the product ''abcd'' is 12.<ref>MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", ''[[Mathematical Gazette]]'' 96, March 2012, pp. 91-96.</ref> The quadruple with the minimal product is (1, 2, 2, 3).

	~~==Relationship with quaternions and rational orthogonal matrices==~~
	A primitive Pythagorean quadruple <math>(a,b,c,d)</math> [[Parametrization\|parametrized]] by <math>(m,n,p,q)</math> corresponds to the first [[Column vector\|column]] of the [[Linear transformation#Matrices\|matrix representation]] <math>E(\alpha)</math> of [[Quaternion#Conjugation.2C_the_norm.2C_and_reciprocal\|conjugation]] <math>\alpha(\cdot)\overline{\alpha}</math> by the [[Hurwitz quaternion]] <math>\alpha = m + ni + pj + qk</math> [[Function (mathematics)#Restrictions and extensions\|restricted]] to the subspace of [[Quaternion\|<math>\mathbb{H}</~~math>]] spanned by <math>i, j, k</math>, which is given by~~

	~~:<math>~~
	~~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},<~~/~~math>~~

	~~where the columns are pairwise [[orthogonality\|orthogonal]] and each has [[Norm_(mathematics)#Euclidean_norm\|norm]] ''d''. Furthermore, we have <math>\frac{1}{d}E(\alpha)<~~/~~math> [[Orthogonal group\|<math>\in \text{SO}(3, \mathbb{Q})<~~/math>]], and, in fact, ''all'' 3 × 3 orthogonal matrices with [[Rational number\|rational]] coefficients arise in this manner.<ref>J. Cremona, ''Letter to the Editor'', Amer. Math. Monthly '''94''' (1987), 757–758.</~~ref>~~

	~~==Pythagorean quadruples with small norm==~~
	: (1,2,2,3), (2,3,6,7), (1,4,8,9), (4,4,7,9), (2,6,9,11), (6,6,7,11), (3,4,12,13), (2,5,14,15), (2, 10, 11, 15), (1,12,12,17), (8,9,12,17), (1,6,18,19), (6,6,17,19), (6,10,15,19), (4,5,20,21), (4,8,19,21), (4,13,16,21), (8,11,16,21), (3,6,22,23), (3,14,18,23), (6,13,18,23), (9, 12, 20, 25), (12, 15, 16, 25), (2,7,26,27), (2,10,25,27), (2,14,23,27), (7,14,22,27), (10,10,23,27), (3,16,24,29), (11,12,24,29), (12,16,21,29)

	~~==See also==~~
	*[[Pythagorean triple]]
	*[[Quaternions and spatial rotation]]
	*[[SO(4)#The Euler–Rodrigues formula for 3D rotations\|Euler-Rodrigues formula for 3D rotations]]
	*[[Euler's sum of powers conjecture]]
	*[[Beal's conjecture]]
	*[[Jacobi–Madden equation]]
	*[[Prouhet–Tarry–Escott problem]]
	*[[Taxicab number]]
	*[[Fermat cubic]]

	~~==References==~~
	~~{{Reflist}}~~

	~~==External links==~~
	*{{MathWorld\|urlname=PythagoreanQuadruple\|title=Pythagorean Quadruple}}
	*{{MathWorld\|urlname=LebesgueIdentity\|title=Lebesgue's Identity}}
	*{{Gutenberg\|no=20073\|name=Diophantine Analysis\|author=Carmichael}}
	*[http://~~www.math~~.~~siu~~.~~edu/kocik/papers/44Cliff.pdf The complete parametrization derived using a Minkowskian Clifford Algebra]~~

	~~[[Category:Additive number theory]]~~
	~~[[Category:Arithmetic problems of plane geometry]]~~
	~~[[Category:Diophantine equations]]~~
	~~[[Category:Diophantine geometry]~~]

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q4367122

2013-03-16T07:13:04Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q4367122

← Older revision		Revision as of 09:13, 16 March 2013
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	~~Friends contact her Claude Gulledge. What she loves doing~~ is ~~to perform croquet but she hasn~~'~~t produced~~ a ~~dime~~ with it. ~~Alabama has usually been his house~~ and ~~his family members loves it~~. ~~His day job~~ is a ~~financial officer but he ideas on changing it~~.<br><br>~~My web page~~ ... ~~car warranty~~ ([~~http:~~//~~racespace~~.~~org~~/~~groups~~/~~auto~~-~~repair~~-~~tips~~-~~make~~-~~your~~-~~car~~-~~running~~-~~smooth~~/ ~~click~~ the ~~next internet page~~])		A '''[[Pythagoras\|Pythagorean]] quadruple''' is a [[tuple]] of [[integer]]s ''a'', ''b'', ''c'' and ''d'', such that ''d > 0'' and <math>a^2 + b^2 + c^2 = d^2</math>, and is often denoted <math>(a,b,c,d)</math>. Geometrically, a Pythagorean quadruple <math>(a,b,c,d)</math> defines a [[cuboid]] with side [[Norm (mathematics)#Euclidean norm\|lengths]] \|''a''\|, \|''b''\|, and \|''c''\|, whose [[space diagonal]] has integer length ''d''. Pythagorean quadruples are thus also called ''Pythagorean Boxes''.<ref>R.A. Beauregard and E. R. Suryanarayan, ''Pythagorean boxes'', Math. Magazine '''74''' (2001), 222–227.</ref>

			== Parametrization of primitive quadruples ==
			The [[Set (mathematics)\|set]] of all '''primitive''' Pythagorean quadruples, i.e., those for which gcd(''a'',''b'',''c'') = 1, where gcd denotes the [[greatest common divisor]] of ''a'', ''b'', and ''c'', is parametrized by,<ref>R.D. Carmichael, ''Diophantine Analysis'', New York: John Wiley & Sons, 1915.</ref><ref>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.</ref><ref>R. Spira, ''The diophantine equation <math>x^2 + y^2 + z^2 = m^2</math>'', Amer. Math. Monthly '''69''' (1962), 360–365.</ref>

			:<math> a = m^2+n^2-p^2-q^2,\,</math>

			:<math> b = 2(mq+np),\,</math>

			:<math> c = 2(nq-mp),\,</math>

			:<math> d = m^2+n^2+p^2+q^2,\,</math>

			where ''m'', ''n'', ''p'', ''q'' are non-negative [[integer]]s and gcd(''m'', ''n'', ''p'', ''q'') = 1 and ''m'' + ''n'' + ''p'' + ''q'' ≡ 1 (mod 2). Thus, all primitive Pythagorean quadruples are characterized by the [[Lebesgue Identity]]

			:<math>(m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2.</math>

			==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 <math>a</math> and <math>b</math> have different parity, let ''p'' be any factor of <math>a^2 + b^2</math> such that <math>p^2 < a^2 + b^2</math>. Then <math>c = (a^2 + b^2 - p^2)/(2p)</math> and <math>d = (a^2 + b^2 + p^2)/(2p)</math>. Note that <math>p = {d - c}</math>.

			A similar method exists<ref>[[Wacław Sierpiński\|Sierpiński, Wacław]], ''Pythagorean Triangles'', Dover, 2003 (orig. 1962), p.102.</ref> for <math>a, b</math> both even, with the further restriction that <math>2p</math> must be an even factor of <math>a^2 + b^2</math>. No such method exists if both ''a'' and ''b'' are odd.

			==Properties==
			The biggest number that always divides the product ''abcd'' is 12.<ref>MacHale, Des, and van den Bosch, Christian, "Generalising a result about Pythagorean triples", ''[[Mathematical Gazette]]'' 96, March 2012, pp. 91-96.</ref> The quadruple with the minimal product is (1, 2, 2, 3).

			==Relationship with quaternions and rational orthogonal matrices==
			A primitive Pythagorean quadruple <math>(a,b,c,d)</math> [[Parametrization\|parametrized]] by <math>(m,n,p,q)</math> corresponds to the first [[Column vector\|column]] of the [[Linear transformation#Matrices\|matrix representation]] <math>E(\alpha)</math> of [[Quaternion#Conjugation.2C_the_norm.2C_and_reciprocal\|conjugation]] <math>\alpha(\cdot)\overline{\alpha}</math> by the [[Hurwitz quaternion]] <math>\alpha = m + ni + pj + qk</math> [[Function (mathematics)#Restrictions and extensions\|restricted]] to the subspace of [[Quaternion\|<math>\mathbb{H}</math>]] spanned by <math>i, j, k</math>, which is given by

			:<math>
			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},</math>

			where the columns are pairwise [[orthogonality\|orthogonal]] and each has [[Norm_(mathematics)#Euclidean_norm\|norm]] ''d''. Furthermore, we have <math>\frac{1}{d}E(\alpha)</math> [[Orthogonal group\|<math>\in \text{SO}(3, \mathbb{Q})</math>]], and, in fact, ''all'' 3 × 3 orthogonal matrices with [[Rational number\|rational]] coefficients arise in this manner.<ref>J. Cremona, ''Letter to the Editor'', Amer. Math. Monthly '''94''' (1987), 757–758.</ref>

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

			==See also==
			*[[Pythagorean triple]]
			*[[Quaternions and spatial rotation]]
			*[[SO(4)#The Euler–Rodrigues formula for 3D rotations\|Euler-Rodrigues formula for 3D rotations]]
			*[[Euler's sum of powers conjecture]]
			*[[Beal's conjecture]]
			*[[Jacobi–Madden equation]]
			*[[Prouhet–Tarry–Escott problem]]
			*[[Taxicab number]]
			*[[Fermat cubic]]

			==References==
			{{Reflist}}

			==External links==
			*{{MathWorld\|urlname=PythagoreanQuadruple\|title=Pythagorean Quadruple}}
			*{{MathWorld\|urlname=LebesgueIdentity\|title=Lebesgue's Identity}}
			*{{Gutenberg\|no=20073\|name=Diophantine Analysis\|author=Carmichael}}
			*[http://www.math.siu.edu/kocik/papers/44Cliff.pdf The complete parametrization derived using a Minkowskian Clifford Algebra]

			[[Category:Additive number theory]]
			[[Category:Arithmetic problems of plane geometry]]
			[[Category:Diophantine equations]]
			[[Category:Diophantine geometry]]

en>Reyk: add illustration

2011-05-20T01:11:20Z

add illustration

New page

Friends contact her Claude Gulledge. What she loves doing is to perform croquet but she hasn't produced a dime with it. Alabama has usually been his house and his family members loves it. His day job is a financial officer but he ideas on changing it.<br><br>My web page ... car warranty ([http://racespace.org/groups/auto-repair-tips-make-your-car-running-smooth/ click the next internet page])