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The abc-conjecture

Establish that for every real number ε>0 there exists an absolute constant C(ε)>0 such that for all triples of coprime integers a, b, c with a+b=c, one has max{|a|,|b|,|c|} ≤ C(ε) · ∏_{p | abc} p^{1+ε}, where the product is over all prime numbers p dividing abc.

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Background

The abc-conjecture, formulated by Masser and Oesterlé, is a central open problem in Diophantine number theory, linking the size of integers in a sum to the product of distinct prime factors (the radical). It has deep connections to many other major conjectures and results, including Szpiro’s conjecture, Vojta’s conjectures, and effective versions of Mordell’s conjecture.

In this paper, the author presents a framework based on Arithmetic Teichmüller Spaces and interprets Mochizuki’s Inter-Universal Teichmüller Theory to outline a proof strategy for the abc-conjecture, situating it among equivalent formulations and reductions via Vojta’s Height Inequality and its strong form.

References

The following tantalizing assertion was conjectured by David Masser and Joseph Oesterle (see ) and is known as the $abc$-conjecture: For each $\varepsilon>0$, there exists an absolute constant $C(\varepsilon)>0$, such that for all primitive triples of $a,b,c$ integers (i.e. triples of integers with $\text{gcd}(a,b,c)=1$) satisfying $$a+b=c,$$ one has $$\max{a},{b},{c}}\leq C(\varepsilon)\cdot \prod_{p|a\cdot b\cdot c}p{1+\varepsilon}$$ where the product is over all the prime number $p$ dividing $a\cdot b\cdot c$.

— Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture (2403.10430 - Joshi, 15 Mar 2024) in Section 2.1 (The $abc$-conjecture)