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ID: INPUT{id.yaml}

Title: >
  $abc$-triples of high merit

Definition: >
  The list contains all known $abc$-triples merit at least $24$, 
  as givin in CITE{ABCatHome}.
  An $abc$-triple is a solution of the equation $a+b=c$ with coprime integers
  $0 < a \leq b < c$.
  The quality of an $abc$-triple is defined as
  $q = \log(c)/\log(r)$, 
  where $r = \text{rad}(abc)$ is the radical of $abc$, that is, 
  the product over all prime divisors of $abc$.
  The merit of an $abc$ triple is defined as 
  $m = (q-1)^2 \log(r) \log\log(r)$.
  
Parameters:
  m:
    type: R
    constraints: $m > 24$
  v:
    title: variable
    type: Symbolic
    show-in-parameter-list: no
   
Comments:
  comment-abc-conjecture: >
    The abc-conjecture CITE{WikiABC} states that $\limsup q = 1$, that is,
    for every $\varepsilon > 0$ there are only finitely many $abc$-triples
    of quality $q > 1+\varepsilon$.
  comment-abc-merit-conjecture: >
    The refined $abc$-conjecture of Robert, Stewart and Tenenbaum CITE{RoStTe14}
    implies that $\limsup m = 48$.
    
Formulas:

Programs:

References:
  RoStTe14:
    bib: >
      O. Robert, C. Stewart, G. Tenenbaum, 
      A refinement of the abc conjecture, 
      Bull. Lond. Math. Soc. 46:6 (2014), 1156-1166.
    url: https://hal.archives-ouvertes.fr/hal-01281526/document
      
Links:
  ABCatHome:
    title: "Bart de Smit: ABC@home" 
    url: https://www.math.leidenuniv.nl/~desmit/abc/
  ABCatHomeHighM:
    title: "Bart de Smit: ABC@home: High merit triples" 
    url:     https://www.math.leidenuniv.nl/~desmit/abc/index.php?set=3
  WikiABC:
    title: "Wikipedia: abc conjecture"
    url: https://en.wikipedia.org/wiki/Abc_conjecture
    
Similar tables:
  
Keywords:
  
Tags:
- number theory
- abc conjecture

Data properties:
  type: Z
  complete: unknown
  sources:
  - CITE{ABCatHomeHighM} and the references therein
  reliability: >
    The given merit $m$ might not be reliable, 
    in which case it is signified with an asterisk '*':
    By definition, $m$ depends on the prime factorization of $abc$,
    which might be out of reach.
    If this is the case, the large remaining factor of $abc$ is
    heuristically assumed to be square-free, 
    such that a heuristic value for $r$ (and thus for $q$ and $m$) can be deduced.

Display properties:
  number-header: value

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$abc$-triples of high merit

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$m$

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Definition

The list contains all known $abc$-triples merit at least $24$, as givin in [2]. An $abc$-triple is a solution of the equation $a+b=c$ with coprime integers $0 < a \leq b < c$. The quality of an $abc$-triple is defined as $q = \log(c)/\log(r)$, where $r = \text{rad}(abc)$ is the radical of $abc$, that is, the product over all prime divisors of $abc$. The merit of an $abc$ triple is defined as $m = (q-1)^2 \log(r) \log\log(r)$.

Parameters

$m$

— real number ($m > 24$)

Comments

(1)

The abc-conjecture [4] states that $\limsup q = 1$, that is, for every $\varepsilon > 0$ there are only finitely many $abc$-triples of quality $q > 1+\varepsilon$.

(2)

The refined $abc$-conjecture of Robert, Stewart and Tenenbaum [1] implies that $\limsup m = 48$.

References

[1]

O. Robert, C. Stewart, G. Tenenbaum, A refinement of the abc conjecture, Bull. Lond. Math. Soc. 46:6 (2014), 1156-1166. https://hal.archives-ouvertes.fr/hal-01281526/document

Links

[2]

Bart de Smit: ABC@home

[3]

Bart de Smit: ABC@home: High merit triples

[4]

Wikipedia: abc conjecture

Data properties

Entries are of type: integer

Table is complete: unknown

Sources of data: [3] and the references therein

Reliability: The given merit $m$ might not be reliable, in which case it is signified with an asterisk '*': By definition, $m$ depends on the prime factorization of $abc$, which might be out of reach. If this is the case, the large remaining factor of $abc$ is heuristically assumed to be square-free, such that a heuristic value for $r$ (and thus for $q$ and $m$) can be deduced.