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Using python I found two solutions: $(4,16,8)$ and $(3,27,9)$. Also $a \neq b \neq c$ and $a,b,c \neq 1$. Are there any more solutions and if not, how to prove it?

Also does a similar equation $a^b + b^a = c^a + c^b$ has any solutions? I couldn't find any for $a,b,c < 100$

CiaPan
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Jan Kuś
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  • So, you want perfect powers (base and exponent greater than $1$) , right ? – Peter Jun 16 '18 at 08:56
  • Such diophantie-equations are already quite complicated. Not sure whether a proof is feasible. If we assume $1<a<b$ and $c>1$ , the equation in the title has no more solutions for $a,b,c\le 300$ and the equation in the body has no solution in the same range. If we assume $a=b$, we have $$2a^a=2a^c$$ which can only hold for $a=c$ and similar for the equation in the body $$2a^a=2c^a$$ also implying $a=c$ – Peter Jun 16 '18 at 09:07
  • I extended the limit to $500$ , no further solutions. – Peter Jun 16 '18 at 09:25

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