# Questions tagged [powerful-numbers]

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4
questions

**33**

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**1**answer

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 = ...

**6**

votes

**0**answers

262 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 ...

**4**

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**0**answers

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### 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

**0**answers

369 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, ...