For $ m=12$ then it has many solution. Prove with the Pell's equation and has discuss .

Can you please prove it has infinitely many for 12?

$ x=a-k,y=a,z=a+k$ Then the equation equivalent: $ a^2-3k^2=1$ It is the Pell equation. The root of equation is : $ a_0=1,k_0=0$ $ a_1=2,k_1=1$ $ a_{n+2}=4a_{n+1}-a_n$ $ k_{n+2}=4k_{n+1}-k_n$ Easy to check $ a>k$ so the equation has positive solution. More consider the equation: $ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{c}{xyz}=\frac{3c+9}{x+y+z}$ With the same method we can prove this equation has infinite natural solution.

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