How to generate Pythagorean triples (example: 3,4,5), well one way at least.
Starting with (x + y)2 = x2 + y2 + 2xy and (x – y)2 = x2 + y2 – 2xy
we can write the difference of two squares
(x + y)2 – (x – y)2 = 4xy
and if we write x = A2 and y = B2 the right hand side is a square as well.
Thus:
(A2 + B2) 2 – (A2 – B2) 2 = 4A2 B2 = (2AB) 2
which can be written as
(A2 + B2) 2 = (A2 – B2) 2 + (2AB) 2
the Pythagoras form.
Now just put in some integers for A and B
2 and 1 gives 3,4,5
Conjecture1: This process generates ALL the Pythagorean triples.
Conjecture2: Every odd number belongs to some Pythagorean triple.
Have fun…….
My next post will be about finding the radius of the inscribed circle in a right angled triangle…..
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On the first line, did you mean y^2 rather than t^2? I can’t figure out where the t^2 came from.
The ‘t’ is a mysterious key right next to the ‘y’.
I have fixed it !
Gracias !
Whew! I thought that I was missing some important mathematical concept that I must have skipped in school! I’m glad it’s just bad typing and not a hole in my mathematics education!