<s>[INST] For integer m greather than 2, $\frac{1}{m} + \frac{1}{m+2}$, the numerators and denomitors are primitive pythagorean triples $a$ and $b$ For $m = 2$, the fraction is $\frac{3}{4}$. for $m=3$, the fraction is$\frac{8}{15}$. I was wondering why numerators and demoninators of  $\frac{1}{m} + \frac{1}{m+2}$ show primitive pythagorean triples a and b.
 [/INST] $$\frac{1}{m} + \frac{1}{m+2} = \frac{2m+2}{m^2 + 2m}$$
Now notice that
$$(2m+2)^2 + (m^2 + 2m)^2=8m^2+4m+4+m^4+4m^3=(m^2+2m+2)^2$$
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