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Inspired by this question, I'd like to know how one would go about proving the below more general equation?

$$n \in \mathbb{N},\;a \in \mathbb{N},\;b \in \mathbb{N}$$ $$b^n+1 \notin a^{\mathbb{N}+1}$$

$$\left\lfloor n\frac{\log (b)}{\log (a)}\right\rfloor =\left\lfloor \frac{\log \left(b^n+1\right)}{\log (a)}\right\rfloor$$

Empirically the above appears, prima facie, to be the case. If it's not, apologies, I tried quite a few values of $a$, $b$ and $n$ and came up with the above assumptions.

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Here's an idea: can you prove $\lfloor \log(b^n)\rfloor = \lfloor \log(b^n+1)\rfloor$? If yes, then you can use $\lfloor x/n\rfloor =\lfloor \lfloor x\rfloor /n\rfloor$ given certain conditions I don't remember exactly at the moment. It shouldn't be too difficult to prove it altogether. –  Ian Mateus Feb 24 at 1:44
Hmm... what about $(a,b,n)=(2,3,1)$? ;-) Maybe you wanted to ask for $b^n+1$ not being a power of $a$? –  Peter Košinár Feb 24 at 2:25
@PeterKošinár Oops, thanks for pointing that out! Fixed. –  Alec Feb 24 at 2:41

1 Answer 1

up vote 2 down vote accepted

Even more is actually true for positive integers $x$ and $a>1$, such that $(x+1)$ is not a power of $a$:

$$\left\lfloor\frac{\log x}{\log a}\right\rfloor = \left\lfloor\frac{\log (x+1)}{\log a}\right\rfloor$$

In order to see why this is so, let $m$ be the largest power of $a$ which doesn't exceed $x$. This means that we have $$a^m \leq x < a^{m+1}$$ and also $$m \leq \log_a x < m + 1$$ Since $\log_a x = \frac{\log x}{\log a}$, we have $$\left\lfloor\frac{\log x}{\log a}\right\rfloor = m$$

The right-hand side floor expression is clearly not smaller than $m$, so we only need to prove it cannot be larger either. Assume (for a proof by contradiction) that this was the case to get $$\frac{\log(x+1)}{\log a} \geq \left\lfloor \frac{\log(x+1)}{\log a}\right\rfloor \geq m+1$$ In the light of inequality $x<a^{m+1}$, the only possibility would be $x+1=a^{m+1}$. But that's precisely what we've forbidden by the not-a-power condition, thus completing our proof.

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Brilliant, thank you very much @Peter! –  Alec Feb 24 at 2:58

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