currently in number theory is there a method to solve for integer solutions to equations like $2^x+3^y=9^z+8^w+8^t$?

For example, $2^{13}+3^6=9^3+8^4+8^4$. (obtained using computer brute force)

In general, is there a way to solve for ${A_1}^{x_1}+{A_2}^{x_2}+\ldots+{A_n}^{x_n}={B_1}^{y_1}+{B_2}^{y_2}+\ldots+{B_m}^{y_n}$ or is it out of scope of the current number theory methods? ($A_i$ and $B_i$ are the given numbers, want to solve for $x_i$ and $y_i$)

Sincere thanks for help!


Your solution is one of a pattern: $$2^{3m+1}+3^{2n}=9^n+8^m+8^m$$ which you can prove with the laws of exponents. This form includes $2^1+3^{2n}=9^n+8^0+8^0$. I would be surprised if there were others, but I have no proof.


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