Number of $p$-smooth $abc$-triples
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Numbers
$p$
$n(p)$
2:
3
3:
21
5:
99
7:
375
11:
1137
13:
3267
17:
8595
19:
21891
23:
52965
29:
120087
31:
267843
37:
572145
41:
1194483
43:
2476743
47:
5037825
53:
9980691
Definition
An $abc$-triple is a solution of the equation $a + b = c$ in (non-zero) coprime integers $a$, $b$, $c$. We call the $abc$-triple $p$-smooth if the largest prime factor of the product $abc$ is at most $p$. This list contains the number $n(p)$ of $p$-smooth $abc$-triples for small values of $p$.
Parameters
$p$
—   integer (prime)
Formulas
(1)
We can mod out a symmetry by requiring $0 < a \leq b < c$. If $n'(p)$ denotes the number of such constrained $p$-smooth $abc$-triples, then $n(p) = 6n'(p) - 3$. This is due to a $6$-fold symmetry (by the anharmonic group [7], which is isomorphic to $S_3$), which is only $3$-fold for the solution $1+1=2$.
Comments
(2)
Let $q = \log\max(|a|,|b|,|c|)/\log\rm{rad}(abc)$ denote the quality of an $abc$-triple. The abc-conjecture [6] states that $\limsup q = 1$, that is, for every $\varepsilon > 0$ there are only finitely many $abc$-triples of quality $q > 1+\varepsilon$.
(3)
A lower bound for $n(p)$ can be deduced from Theorem 4 of [1].
(4)
An upper bound for $n(p)$ can be deduced from Theorem 1 of [2].
References
[1]
P. Erdös; C. L. Steward; R. Tijdeman, Compositio Mathematica 66:1 (1988), page 37-56. https://eudml.org/doc/89898
[2]
J. H. Evertse, "On equations in S-units and the Thue-Mahler equation". Invent. math. 75 (1984), 561-584.
[3]
R. von Känel, B. Matschke, "Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture". 2016 (github)
Links
Data properties
Entries are of type: real number
Table is complete: no
Sources of data: [3] (data: [5])
Reliability: provably correct