Good evening.

I'm struggling to understand an inequality on a solution proposed to this problem :

For each positive integer $n$ there exist $n$ consecutive positive integers none of which is an integral power of a prime number.

What I don't understand in the solution :

There are at most : $$1+\sqrt{n} +\sqrt[3]{n} + \sqrt[4]{n} + \dots + \sqrt[E(\log_2(n))]{n} $$ "true" powers of the form $m^k$, $k\geqslant 2$ in the set $\{1, \dots, n\}$.

(If $x \in \mathbb{R}$, $E(x)$ is the integer part of $x$)

Where does that come from ? Why can we stop at $\sqrt[E(\log_2(n))]{n}$ ?

I've never saw that kind of majoration.


1 Answer 1


Let $m\ne 1$. Then $m\geq 2$ and so $2^k\leq m^k$.

But we know that $m^k\leq n$. So $2^k\leq m^k\leq n$. This implies $k\leq \log_2(n)$.


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