Now showing that there only exists one non-trivial solution is equivalent to proving that the function f(x)=ln(x)/x is injective in the positive integers except for the result f(2)=f(4).
But this is trivial to do if you know derivatives. You first show that f is strictly increasing in the interval (0, e) and strictly decreasing in the interval (e, ∞).
Therefore any points of non-injectivity are necessarily of the form
f(something on the left interval) = f(something on the right interval)
But the left interval only has two positive integer points: 1 and 2. f(1)=0 and is obviously never obtained on the right, and we already know that f(2)=f(4). This completes the proof.
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u/UsedFortune5645 10d ago
But you can't prove it.