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Results tagged with abc-conjecture
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user 2480
3
votes
Is this set dense in [0,+∞)?
This is not a full answer, but a pair of soft arguments suggesting that $A$ is dense in $[0, +\infty)$.
First Argument
Given any triple $(a,b,c)$, let $\displaystyle r(a,b,c)=\frac{c}{\text{rad}(abc) …
- 5,103
10
votes
0
answers
174
views
Is almost every number the sum of two numbers with small radicals?
Define a set of numbers with small radicals (A341645 in OEIS) by
$$A_2=\{n\in\mathbb{N} \;|\; \text{rad}(n)^2\le n\}$$
The asymptotic density of $A_2\cap \{1,\dots N\}$ is $\sqrt{N}\times e^{2(1+o(1)) …
- 5,103
10
votes
0
answers
465
views
Mini-$abc$ conjecture
Define $\text{rad}_{23}(2^m3^nr)=2^{\text{sign}(m)}3^{\text{sign}(n)}r$, where $m,n\ge0$ and $2,3\nmid r\in\mathbb{N}$.
For a triple $a+b=c$ define the quality $q_{23}(a,b,c)=\frac{\log(c)}{\log(\tex …
- 5,103