<s>[INST] How many solutions to each of the equations $2^x+3^y=5^z$, $2^x+5^y=3^z$, $3^x+5^y=2^z?$ Let $x,y,z\in\Bbb{N}$ How many total solutions are there to each of the three distinct equations below?
$$2^x+3^y=5^z \tag 1$$
$$2^x+5^y=3^z \tag 2$$
$$3^x+5^y=2^z \tag 3$$
I found 3 solutions to equation (1), 4 solutions to equation (2), 5 solutions to equation (3),(12 altogether) which might be all of them.
$$3^0+5^0=2^1$$
$$5^0+2^1=3^1$$
$$5^0+3^1=2^2$$
$$3^0+2^2=5^1$$
$$2^1+3^1=5^1$$
$$3^1+5^1=2^3$$
$$5^0+2^3=3^2$$
$$2^2+5^1=3^2$$
$$3^2+2^4=5^2$$
$$2^1+5^2=3^3$$
$$5^1+3^3=2^5$$
$$3^1+5^3=2^7$$
 [/INST] Mahler proved this equation has only finite solutions.$a^x+b^y=c^z$
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