Questions tagged [powerful-numbers]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
0answers
58 views

What's equal the below power nested radical?

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}{\...
6
votes
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 ...
33
votes
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 = ...
4
votes
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 ...
2
votes
0answers
367 views

Counting factors: is this approach in the literature on multiperfect numbers?

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, ...