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	Pythagorean Triple Calculator in JavaScript
		If you want to ensure that the triple can't be reduced by dividing by a common factor, follow this rule: make s and t coprime (having no common divisors other than 1) and not both odd.
		A non-reducable Pythagorean triple is called a primitive Pythagorean triple.
		It can easily be verified that x^2 +y^2 = z^2, with x, y, and z in terms of s and t, the tough part is in deriving these equations from scratch.
	www.math.clemson.edu /~simms/neat/math/pyth (374 words)

	NationMaster - Encyclopedia: Pythagorean triple
		A primitive Pythagorean triple is one in which a, b and c are coprime.
		The triangles described by non-primitive Pythagorean triples are always similar to the triangle described by a smaller primitive Pythagorean triple.
		It is interesting to note that there are more than one primitive Pythagorean triple with the same lowest integer, the first example is for 20, which is the lowest integer of two primitive triples: 20 21 29 and 20 99 101.
	www.nationmaster.com /encyclopedia/Pythagorean-triple (464 words)

	Kids.Net.Au - Encyclopedia > Pythagorean triple
		The name comes from the Pythagorean Theorem, which states that any right triangle with integer side lengths yields a Pythagorean triple.
		A Pythagorean triple is said to be primitive if a, b and c have no common divisor.
		Fermat's Last Theorem states that non-trivial triples analogous to Pythagorean triples but with exponents higher than 2 don't exist.
	www.kids.net.au /encyclopedia-wiki/py/Pythagorean_triple (237 words)

	Pythagorean Triples
		A Pythagorean triple which is not a multiple of a smaller one is called a primitive Pythagorean triple.
		This is a Pythagorean triple since, as a triangle, is it just 3 times the 3-4-5 triangle (by which we mean that we just triple the lengths of each side of a 3-4-5 triangle, which we already know is right-angled).
		Pythagorean Triples Projects is Eric Rowland's useful page of further ideas for your own investigations together with some hints and solutions.
	www.mcs.surrey.ac.uk /Personal/R.Knott/Pythag/pythag.html (7858 words)

	Pythagorean Triples
		Although the Pythagorean theorem arose in geometry, we will be concerned strictly with the number theoretic properties of the Pythagorean equation, using the connection to geometry only as a jumping off point.
		In general, the number of primitive Pythagorean triples of hypotenuse n is dependent on the number of prime factors of n that are congruent to 1 modulo 4.
		Sometimes a triple that has a pair of sides with only a unit difference is referred to as a "twin." For example, the triples (3, 4, 5), (5, 12, 13), (7, 24, 25), and (21, 20, 29) each have a pair of sides that differ by 1.
	www.math.rutgers.edu /~erowland/pythagoreantriples.html (3628 words)

	math lessons - Pythagorean triple
		A primitive Pythagorean triple is one in which a, b and c are coprime.
		The triangles described by non-primitive Pythagorean triples are always similar to the triangle described by a smaller primitive Pythagorean triple.
		It is interesting to note that there are more than one primitive Pythagorean triple with the same lowest integer, the first example is for 20, which is the lowest integer of two primitive triples: 20 21 29 and 20 99 101.
	www.mathdaily.com /lessons/Pythagorean_triple (470 words)

	PlanetMath: Pythagorean triplet
		It follows that there are countably many Pythagorean triplets.
		All the primitive Pythagorean triplets are given by
		This is version 6 of Pythagorean triplet, born on 2001-10-06, modified 2005-12-15.
	planetmath.org /encyclopedia/PythagoreanTriple2.html (116 words)

	Pythagorean theorem: Definition and Links by Encyclopedian.com
		The Pythagorean theorem or Pythagoras' theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, though the facts of the theorem were known before he lived.
		This can be proven using the law of cosines which is a generalization of the Pythagorean theorem applying to all triangles, not just right-angled ones.
		One should note that the Pythagorean Theorem is derived from the axioms of Euclidean geometry and does not apply to triangles in non-Euclidean spaces such as the surface of a sphere.
	www.encyclopedian.com /py/Pythagorean-Theorem.html (808 words)

	Pythagorean triple - Wikipedia, the free encyclopedia
		A primitive Pythagorean triple is one in which a, b and c are coprime.
		The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution.
		In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers.
	en.wikipedia.org /wiki/Pythagorean_triple (2160 words)

	Pythagorean Theorems - Some 'Not So Familiar' Implications
		Still, the Pythagorean relationship holds, the sum of the areas of the rectangles drawn on the two legs is equal to the area of the rhombus drawn on the hypotenuse of the right triangle.
		In these figures the Pythagorean relationship still holds, the sum of the areas of the parallelograms drawn on the two sides is equal to the area of the parallelogram drawn on the third side of the triangle.
		Similar to Pythagorean Triples, quadruples are sets of 4 integers, such that the sum of the squares of the smaller three equals the square of the fourth larger integer.
	www.contracosta.edu /math/pythagoras.htm (1637 words)

	Pythagorean theorem - ExampleProblems.com
		The Pythagorean theorem: The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse.
		A paraphrase of the Pythagorean theorem is :
		The Pythagorean Theorem is Equivalent to the Parallel Postulate.
	www.exampleproblems.com /wiki/index.php/Pythagorean_theorem (1767 words)

	Pythagorean theorem Summary
		The Pythagoreans formulated a view from an arithmetical standpoint that believed the concept of the number was the key to the qualities of mankind and matter and that it was the ultimate principle of all proportion of the universe.
		The Pythagoreans proved that the square root of two is irrational, and this proof has come down to us even though it flew in the face of their cherished belief that everything was rational.
		Another generalization of the Pythagorean theorem to three dimensions is de Gua's theorem: If a tetrahedron has a right angle corner (a corner like a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces.
	www.bookrags.com /Pythagorean_theorem (4628 words)

	Learn more about Pythagorean theorem in the online encyclopedia. (Site not responding. Last check: )
		The Pythagorean theorem or Pythagoras' theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, though the facts of the theorem were known before he lived.
		The Pythagorean theorem stated in Cartesian coordinates is the formula for the distance between points in the plane -- if (a, b) and (c, d) are points in the plane, then the distance between them is given by
		Since the Pythagorean theorem is derived from the axioms of Euclidean geometry, and physical space may not always be Euclidean, it need not be true of triangles in physical space.
	www.onlineencyclopedia.org /p/py/pythagorean_theorem.html (926 words)

	Math Forum: Ask Dr. Math FAQ: Pythagorean Triples
		Pythagorean triples with this property are called primitive.
		Pythagorean Triples (Sums of 3 cubes equal a cube; sums of 4th powers equal a 4th power)
		Pythagorean Triples (One of (a,b,c) is divisible by 3, one by 4, and one by 5)
	mathforum.org /dr.math/faq/faq.pythag.triples.html (780 words)

	[No title] (Site not responding. Last check: )
		I wrote down a table of Pythagorean triples and asked the students to find the pattern: 3^2 + 4^2 = 5^2 8^2 + 6^2 = 10^2 15^2 + 8^2 = 17^2 24^2 + 10^2 = 26^2 35^2 + 12^2 = 37^2....
		Thus, the triple 3,4,5 is primitive but the triple 8,6,10 is not.
		(E.g., the 3,4,5 triple gives rise to the rational point (.6,.8) on the unit circle.) Conversely, if we have a rational point on the circle, it must be of the form ((1-t^2)/(1+t^2),2t/(1+t^2)) for some rational number t; writing t = p/q, and then clearing the denominator, we get the Pythagorean triple q^2-p^2, 2pq, q^2+p^2.
	www.math.wisc.edu /~propp/courses/491/9.02 (755 words)

	Pythagorean Activity
		For a Pythagorean triple, it is true that there are either two odd side lengths or no odd side lengths.
		The Pythagorean Theorem is named after Pythagoras and his school who are thought to have given the first type of proof of the famous theorem.
		Whether the Pythagoreans actual gave a modern and correct proof of the theorem remains unknown, but their rigor with numbers leads us to suggest they may have produced something like a demonstration that satisfied the level of rigor of the day.
	distance-ed.math.tamu.edu /pasadena/measurement/pythagorean_thm/pythagorean_puzzle.html (1279 words)

	Pythagorean Triples
		Pythagorean Triples are a set of three integer numbers that satisfy the Pythagorean equation.
		The Pythagorean triple property for a set of integer numbers will hold when all three numbers are multiplied by the same number.
		Hence, to find the Pythagorean triple for the smallest number 2a being even, find the triple for a, an odd number, and multiply its triple by 2.
	library.thinkquest.org /C0110248/trigonometry/pythagoras3.htm (264 words)

	Pythagorean Theorem (Site not responding. Last check: )
		The Pythagorean Theorem states that the sum of the areas of the squares on the two legs of a right triangle is equal to the area of a square on the hypotenuse (the side opposite the 90º or right angle).
		This is known as a Pythagorean triple: all the sides have lengths which are whole numbers.
		There are infinitely many Pythagorean triples, such that the sides of a right triangle are whole numbers.
	www.algebralab.org /lessons/lesson.aspx?file=Trigonometry_TrigPythagoreanThm.xml (653 words)

	Pythagorean triple - Pythagorean Triples A pythagorean triple is a triple of integers a (Site not responding. Last check: )
		Pythagorean triple - Pythagorean Triples A pythagorean triple is a triple of integers a
		A Pythagorean triple consists of three positive integers a, b, and c, A primitive Pythagorean triple is one in which a, b and c are coprime.
		Pythagorean Triples A pythagorean triple is a triple of integers a
	infomemory.com /ifmm/pythagorean-triple.htm (247 words)

	The Prime Glossary: Pythagorean triples
		Integer triples which satisfy this equation are Pythagorean triples.
		The triples for which the entries are relativey prime are called primitive.
		Hopefully you answered 'no.' In any primitive Pythagorean triple one of the three entries must be even, and it is easy to show that 2 can not be the side of a Pythagorean triple (look modulo 8).
	primes.utm.edu /glossary/page.php?sort=PrmPythagTriples (336 words)

	Primitive Pythagorean Triples
		Pythagorean Triple is a triple of positive integers a,b,c such that a
		Primitive Pythagorean Triple is a Pythagorean triple a,b,c with the constraint that gcd(a,b)=1, which implies gcd(a,c)=1 and gcd(b,c)=1.
		Let a,b,c be a primitive Pythagorean triple, a denote the odd leg, b denote the even leg, c denote the odd hypotenuse.
	f2.org /maths/ppt.html (638 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: )
		If you want to find all Pythagorean triples, it suffices to find all "primitive" Pythagorean triples, that is, ones such that the set {a,b,c} has greatest common divisor 1.
		The formulas you give are the ones which give primitive Pythagorean triples, provided appropriate restrictions are placed on m and n.
		Thus the formulas that give all Pythagorean triples are these: a = d(m^2 - n^2), b = 2dmn, c = d*(m^2 + n^2), where d is any positive integer, m > n > 0 are integers of opposite parity and relatively prime.
	mathforum.org /library/drmath/view/55811.html (1198 words)

	William H. Richardson -- Pythagorean Triples
		For example, all the triples for which s = 1 we have that a and c are consecutive odd integers.
		Furthermore, the triples [3 4 5] and [5 12 13] have b and c as consecutive integers.
		We therefore have a generator for the triples with a and b consecutive integers.
	www.math.wichita.edu /~richardson/pythagoreantriples.html (834 words)

	Pythagorean Triples Project
		The Pythagorean theorem is named for the Greek mathematician Pythagoras, who lived in the 6th century BCE, though the theorem had been known elsewhere for some time before.
		Theorem 1 (Pythagorean Theorem and converse) Let x, y, and z be positive numbers.
		Use this to find numbers which are the hypotenuses of many (at least five) primitive Pythagorean triples.
	www.math.rutgers.edu /~erowland/pythagoreantriples-project.html (883 words)

	Pythagorean Puzzle
		The first element (a leg of the corresponding triangle) of each primitive Pythagorean triple is odd; the second (another leg) is a multiple of 4; and the hypotenuse is odd.
		The primitive Pythagorean triple from which our answers are derived must be an odd factor of 42, which means it is 3, 7, or 21.
		This is because r and s have opposite parity, so their difference and their sum can't be even, so no primitive Pythagorean triple has 2 as a factor of its first element, and the second element always has two twice as a factor.
	mcraefamily.com /MathHelp/PythagTriplesWithProductOfPrimesLeg.htm (1847 words)

	Ivars Peterson's MathTrek -Square of the Hypotenuse
		The Pythagorean theorem "is probably the only nontrivial theorem in mathematics that most people know by heart," comments Darko Veljan of the University of Zagreb in Croatia.
		In general, for any number k, the corresponding Pythagorean triple is a = 2k + 1, b = 2k(k + 1), and c = b + 1.
		Generalizing the Pythagorean equation for triangles with integer sides to powers greater than 2 leads to Fermat's last theorem and the so-called ABC conjecture (see The Amazing ABC Conjecture, December 8, 1997).
	www.maa.org /mathland/mathtrek_11_27_00.html (993 words)

	Karl's Calculus Tutor: Rational Unit Circle Points/Pythagorean Triples
		Such triples of integers are known as Pythagorean triples or sometimes as Pythagorean triplets.
		But notice that if you have two Pythagorean triples, and one is multiple of the other, then they both correspond to the same unit circle point.
		You should be able to see that each distinct such Pythagorean triple (that is each triple with no common factors) corresponds to a unique point on the unit circle.
	www.karlscalculus.org /pythtrip.html (1305 words)

	Pythagorean Triples
		There is a simple formula that gives all the Pythagorean triples.
		It's easy to check algebraically that the sum of the squares of the first two is the same as the square of the last one.
		Here are the first few triples for m and n between 1 and 10.
	www.math.uic.edu /~fields/puzzle/triples.html (114 words)

	Pythagorean triple and the dimensions of the Khufu pyramid
		Pythagorean triple and the dimensions of the Khufu pyramid
		A Pythagorean triple is a set of three positive whole numbers a, b, and c that are the lengths of the sides of a right triangle.
		Remark: there is no Pythagorean triple that approximates the angle of 51.84° of the Khufu (Cheops) Pyramid (see value for angle β).
	www.cheops-pyramide.ch /khufu-pyramid/pythagorean-triple.html (101 words)

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