informal introduction to the problem is: does in the set of all powers exist arbitrarily long "spaces" not covered by any power?

Strict formulation of the problem is:

let \(\displaystyle p(n):=\min\{|n-a^k|; a,k \in \mathbb{N}, k>1\}\). Then the question is:

Is \(\displaystyle \sup_{n \in N}\{t(n)\}\) finite?

In other words - having set \(\displaystyle M:=\{a^k; a,k \in \mathbb{N}, k>1\}\). Does then for every natural b exist c such that \(\displaystyle M \cap \{c,c+1,c+2,..,c+b-1\}=\emptyset\)?

Possible variations of the problem are:

- we allow "a" to be only prime numbers

- we allow e.g k>2 (this is shape of numbers in Fermat's last theorem)

- instead of a^k we will examine another function...

- ...

Than you for any ideas for proving or disproving this problem (and even the variations).