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Mathematics-Online course: Linear Algebra - Basic Structures - Scalar Product and Norm

Pythagorean Triple

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A Pythagorean triplet consists of three natural numbers $ l,m,n$ with

$\displaystyle l^2 + m^2 = n^2\, , $

for example: $ (3,4,5)$.

Hence, a triangle for which the lengths of its sides are in the same ratios as the numbers of a Pythagorean triplet is a rectangular triangle.

The Egyptians are said to have used this to obtain right angles: They took a rope devided by knots into twelve segments of equal length. Forming a triangle with sides of three, four and five segments, they obtained a right angle between the two smaller sides.

Any odd number $ l \geq 3$ can be completed to a Pythegorean triplet by the numbers $ m=(l^2-1)/2$ and $ n=(l^2+1)/2$, because we have

$\displaystyle l^2+\frac{l^4-2l^2+1}{4}=\frac{l^4+2l^2+1}{4} =\left(\frac{l^2+1}{2}\right)^2\,. $

Multiplying these triplets by $ 2$ we obtain triplets for any even number $ l \geq 6$.

The given construction principle is a special case of a method resulting from the binomial formula, because from

$\displaystyle c^2-b^2=(c-b)(c+b)=a^2 $

we obtain an integer solution, if we can decompose $ a^2$ into two different factors the difference of which is even. For odd $ a$ this condition is satisfied for the decomposition $ a^2=a^2\cdot 1$.

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005