2
votes
Find all positive integer $n$ such that $\tau(n) = \sqrt{n+1}$
It's easy to prove by induction that $k+1\leq 3^{k/2}$ for all $k\neq 1$ (note that for $k=1$ we have $2<\sqrt 3$) and that $k+1\leq 5^{k/2}$ for all $k$. So if $\nu_3(n)\neq 1$, remembering that $...
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1
vote
Find all positive integer $n$ such that $\tau(n) = \sqrt{n+1}$
A simple method is to use the fact that $\tau(n)$ grows more slowly than any power of $n$. Various explicit estimates of this type exist, which will reduce the problem to a finite computation. For ...
- 85.1k
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