# Questions tagged [powerful-numbers]

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The same copy of this question is montioned here in SE with no convinced Answer , I want to know what MO will say about the below nested radical as a power form it is well known that $$\frac{2}{\... 0answers 260 views ### Is every powerful number the sum of a powerful number and a prime? A positive integer n is called powerful (OEIS: A001694) if p^2 divides n whenever p divides n. Equivalently, n is powerful if n = a^2b^3, where a and b are positive integers. By ... 1answer 2k views ### x^4+y^4 powerful for relatively prime x,y I asked this question on the NMBRTHRY mailing list on 17 February 2014, but it remains unsolved as far as I know. Recall that a "powerful number" is a positive integer whose prime factorizations m = ... 0answers 355 views ### Integer solutions of  z^3 y^2 = x(x-1)(x+1) According to a conjecture there are no three consecutive powerful numbers. Necessary condition for this is integer solution of$$ z^3 y^2 = x(x-1)(x+1) \qquad (1)  What are integer solutions of ...
Does the following approach (or something near it) exist in the number theory literature? I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$ and for this question. First, ...