Oesterlé-Masser Conjecture
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
- 1985: D.W. Masser: Open problems (Proceedings of the Symposium on Analytic Number Theory) (edited by W.W.L. Chen)
- 1988: Joseph Oesterlé: Nouvelles approches du "théorème" de Fermat", Séminaire Bourbaki exp 694 (Astérisque Vol. 161: pp. 165 – 186)
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): $abc$ conjecture