A question I really like because of the logical way of thinking about the answer. First we assume that we do not allow the trivial answer where $$n=m=q=0$$. Then rewrite the equation as: $2^n + 3^m = 5^q+1.$ Now $$2^n$$ must be even for all values of $$n$$, and $$3^m$$ must be odd for all values of $$m$$. Thus $$2^n+3^m$$ is odd for all values of $$n$$ and $$m$$. Similarly, $$5^q$$ is odd for all values of $$q$$, thus $$5^q+1$$ must be even. Therefore the left hand side of the rewritten equation must be odd, whilst the right hand side must be even. Since this is not possible there are no positive integer solutions.

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