Number of $p$-smooth $abc$-triples

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abc conjecture
number theory
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