O Geometry Pythagorean Triples per Sierpinski page 8 10 31

Pythagorean Triples per Sierpinski page 8and Proof of Right Triangle Area = xy “Every primitive Pythagorean triple a, b, c where b is an even integer is obtained only once per the foll

Pythagorean Triples per Sierpinski page 8

and Proof of Right Triangle Area = xy

“Every primitive Pythagorean triple (a, b, c) where b is an even integer

is obtained only once per the following, where u and v are odd and

u v :

a uv b  c 

Here, u and v represent all pairs of odd, relatively prime natural

numbers.” Waclaw Sierpinski, Pythagorean Triangles, The Scripta

Mathematics Studies Number Nine New York City: Graduate School of

Science, Yeshiva University, 1962

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Proof of Right Triangle Area = xy

Dr Stan Hartzler Archer City High School

x

x

y

r r

r y

Area = 1

2r perimeter = 1 (2 2 2 ) ( )

2r r x y r r x y 

2

2

x r y r x y

x xr r y yr r x xy y

xr r yr r xy

xr r yr xy

xr r yr xy

r x r y xy

Fun Relationships

( )

x xy y x y

a b c v u v

r         

2

2

2 2

v uv

x a r

u uv

y b r



  



  

6 2 6 2

x xy y x y

a x r

x xy y x y

b y r

c x y

  

  

 

Perimeter = a b c  2(r x y  )  x y x2 6xy y 2 uv u 2

ab u v uv

xy 

 

Developing u, v for a given r

Example: r = 3 × 5 × 7 = 105

2

uv v v r

v

v



The number of distinct values of (u, v) for a given r of n odd prime

factors is 2n And note that uk  vk  u2 1n k  v2 1n k