<s>[INST] Can two pythagoras triplet have a common number If I have a pythagoras triplet $(a,b,c)$ such that 
$$a^2+b^2=c^2$$
then is there another triplet $(a,d,e)$ possible such that
$$a^2+d^2=e^2, \; b\neq d$$
 [/INST] Each of $\quad A,B,C\quad$ will have as many primitive triples as its number of unique prime factors.
Triples are often generated with Euclid's formula
$$\text{where} \quad A=m^2-k^2 \quad B=2mk \quad C=m^2+k^2$$
$A=3\times5=15\longrightarrow 
f(4,1)=(15,8,17)\quad 
f(8,7)=(15,112,113)$
$B=2^2\times3=12\longrightarrow 
f(3,2)=(5,12,13)\quad 
f(6,1)=(35,12,37)$
$C=5\times13=65\longrightarrow 
f(7,4)=(33,56,65)\quad 
f(8,1)=(63,16,65)$
There are no tiple pairs where two numbers a shared so there is no such thing as
$\quad a^2+b^2=c^2\quad a^2+d^2=c^2\quad b\ne d$
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