Fractions that generate Pythagorean Triples | Colin

An interesting tweet, some time ago, from @RJS2212:
Two unit fractions where denominators differ by 2
Add fractions
Numerator & denominator of the sum are two smaller numbers of Pythag triple

— Robert J Smith (@RJS2212) May 30, 2016

And of course, you wonder two things: a) why does it work, and b) can all Pythagorean triples be written that way?

The first one is an easier proposition, and I’ll set it up like so: we have $\frac{1}{n-1} + \frac{1}{n+1} = \frac{2n}{n^2-1}$.

Are these the legs of a Pythagorean triangle? Let’s see, is $(2n)^2 + (n^2 – 1)^2$ a square? It expands as…

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