<s>[INST] Three primitive Pythagorean triples with the same c Can someone give me an example of Three primitive Pythagorean triples with the same c?
 [/INST] We produce an answer with hypotenuse $c=5\cdot 13\cdot 17$. So we need to find three relatively prime pairs $\{m,n\}$ of opposite parity such that $m^2+n^2=5\cdot 13\cdot 17$. 
It is simplest to calculate using complex numbers, although the same goal can be achieved by using the Brahmagupta Identity.  Note that $c$ factors over the Gaussian integers as $(2+i)(2-i)(3+2i)(3-2i)(4+i)(4-i)$. 
First one: Note that $(2+i)(3+2i)(4+i)=9+32i$. That gives $m=9$, $n=32$. 
Second one: Do the same calculation using $(2-i)(3+2i)(4+i)$.
Third one: Same calculation, using $(2+i)(3-2i)(4+i)$. 
Remark: There is a fourth, using $(2+i)(3+i)(4-i)$. For an example with $8$ instead of $4$, we can play the same game with $5\cdot 13\cdot 17\cdot 29$. 
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