Oesterlé-Masser Conjecture

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Conjecture

Let $\epsilon \in \R$ be a strictly positive real number.

Formulation 1

There exists only a finite number of triples of (strictly) positive integers $\tuple {a, b, c}$ with the conditions:

$a + b = c$
$a$, $b$ and $c$ are pairwise coprime

such that:

$c > \map \Rad {a b c}^{1 + \epsilon}$

where $\Rad$ denotes the radical of an integer.


Formulation 2

There exists a constant $K_\epsilon$ such that for all triples of (strictly) positive integers $\tuple {a, b, c}$ with the conditions:

$a + b = c$
$a$, $b$ and $c$ are pairwise coprime

such that:

$c < K_\epsilon \map \Rad {a b c}^{1 + \epsilon}$

where $\Rad$ denotes the radical of an integer.


Formulation 3

There exists only a finite number of triples of (strictly) positive integers $\tuple {a, b, c}$ with the conditions:

$a + b = c$
$a$, $b$ and $c$ are pairwise coprime

such that:

$\map q {a, b, c} > 1 + \epsilon$

where $\map q {a, b, c}$ denotes the quality of $\tuple {a, b, c}$.


Also known as

The Oesterlé-Masser conjecture is also (more commonly) referred to as the $abc$ conjecture.


Also see


Source of Name

This entry was named for Joseph Oesterlé‎ and David William Masser.


Historical Note

The Oesterlé-Masser Conjecture was first proposed by David William Masser in $1985$ and Joseph Oesterlé in $1988$.

In $2012$, Shinichi Mochizuki published a series of papers which claimed to have proved it.

However, there is perceived to be a gap in this proof.

Despite the misgivings of a number of mathematicians, it was announced on $3$rd April $2020$ that Mochizuki's proof would be published in a journal of which Mochizuki himself is the chief editor.

In the face of all this, it is still generally recognised in the mathematical community that the Oesterlé-Masser Conjecture remains unproven.


Sources