Questions tagged [powerful-numbers]

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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 $p$ is a prime that divides $n$. Equivalently, $n$ is powerful if $n = a^2b^3$, where $a$ and $b$ are positive ...
Pietro Paparella's user avatar
4 votes
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Can $z^3+1$ be powerful for integer $z$ other than $-1,0,2$ and $23$?

In 1976, Schinzel and Tijdeman proved that if a polynomial $P(z)$ with integer coefficients has at least $3$ simple zeros, then there may be at most finitely many $z$ such that $P(z)$ is a perfect ...
Bogdan Grechuk's user avatar
2 votes
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Baby $abc$ conjecture for $n$-th roots

Is there any progress on a “baby $abc$ conjecture” where you restrict attention to rational approximations of $n$-th roots? Let $r/s$ be a very close approximation to $(t/u)^{1/n}$, so that $$ |u\cdot ...
Joe Shipman's user avatar
2 votes
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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, ...
The Masked Avenger's user avatar