Proofreading of a short passage (number theory)

Question

So, I have the following expression: $$(-5)\cdot{3}^m+(-3)\cdot{5}^m=(8k)\cdot3^m+(8k)\cdot5^m$$ ,where $k $ is an integer. I arrived at this by seeking for a contraddiction in a proof. Can I just equate the coefficients of the $3^m$ and $5^m$ terms and claim: $$8k=(-5) \;\;\;;\;\; 8k=(-3)$$ ? This is clearly impossible, otherwise also I know that $k$ must be negative so if I just move things around: $$-(3^m)(8k+5)=5^m(8k+3)$$ Then $(8k+5)$ and $(8k+3)$ would have opposite signs which once again, for integer $k$, it's trivially impossible. Are both these arguements valid? If they aren't, why?

Answer 1

This is too long for a comment. So I'll post it as an answer, though technically it is not.

Simply equating coefficients does not work here. For example $$3^0+5^0=2.3^0+0.5^0$$ Because both sets of the form $\{3^n\}_{n\in\mathbb{N}}$ and $\{5^n\}_{n\in\mathbb{N}}$ spans $\mathbb{N}=\{0,1,2,\cdots\}$ over $\mathbb{N}.$ It means you can write any integer as a finite linear combination of either elements of first set or the second set. These combinations (expansions) are unique if you impose the conditions that coefficients for first set are $\mathbb{Z}_3=\{0,1,2\}$ and coefficient for the second set are $\mathbb{Z}_5=\{0,1,2,3,4\}.$ For example $$123=3^4+3^3+3^2+2.3^1=4.5^2+4.5^1+3.5^0$$ These are just base $3$ and base $5$ expansions of $123 :)$