Primitive Pythagorean triangles with sides of certain forms (Q6542785)

In this paper under review, the authors show that there exist infinitely many primitive Pythagorean triangles such that each side is expressed as a sum of two squares or of cubes.\NMore precisely, let\N\begin{align*}\Nx&=(d^2-2+k^2)^2,\quad y=4|d(d^2-2+k^2)|,\\\Np&=4|d(d^2+2-k^2)|,\quad q=8|k|d^2,\\\Nr&=d^4+2(k^2+2)d^2+(k^2-2)^2,\quad s=8d^2,\N\end{align*}\Nwhere \(d\) and \(k\) are integers satisfying \(d \not \equiv k \pmod2\), \(\gcd(k^2-2,d)=1\) and \(dk \ne0\).\NThen, \(x,y,p,q,r,s\) are positive integers such that \(X=x^2+y^2\), \(Y=p^2+q^2\) and \(Z=r^2+s^2\) satisfy \(X^2+Y^2=Z^2\) with \(\gcd(X,Y,Z)=1\).\NNote that putting \(k=0\) yields\N\begin{align*}\Nx&=(d^2-2)^2,\quad y=4|d(d^2-2)|,\\\Np&=4|d|(d^2+2),\quad q=0,\\\Nr&=d^4+4d^2+4,\quad s=8d^2,\N\end{align*}\Nin which case \(X=x^2+y^2\), \(Y=p^2\) and \(Z=r^2+s^2\) are primitive if \(d\) is odd.\NSimilarly, let\N\begin{align*}\Nx&=9a^4-3a,\quad y=9a^3-1,\\\Np&=q=3a^2,\quad r=9a^4,\quad s=1,\N\end{align*}\Nwhere \(a\) is an even positive integer.\NThen, \(X=x^3+y^3=729a^{12}-1\), \(Y=p^3+q^3=54a^6\), \(Z=r^3+s^3=729a^{12}+1\) satisfy \(X^2+Y^2=Z^2\) with \(\gcd(X,Y,Z)=1\).\N\NThe authors further construct infinitely many rational right triangles \((X,Y,Z)\) of the forms \(X=1/x+1/y\), \(Y=1/p+1/q\), \(Z=1/r+1/s\) and \(X=1/x-1/y\), \(Y=1/p-1/q\), \(Z=1/r-1/s\), respectively, where \(x,y,p,q,r,s\) are positive integers.