# ABC Conjecture has a serious claim to a proof

Wikipedia – The abc conjecture (also known as Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985) as an integer analogue of the Mason–Stothers theorem for polynomials. The conjecture is stated in terms of three positive integers, a, b and c (whence comes the name), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is rarely much smaller than c.

The abc conjecture has already become well known for the number of interesting consequences it entails. Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture. Goldfeld (1996) described the abc conjecture as “the most important unsolved problem in Diophantine analysis”.

In August 2012, Shinichi Mochizuki released a paper with a serious claim to a proof of the abc conjecture. Mochizuki calls the theory on which this proof is based inter-universal Teichmüller theory, and it has other applications including a proof of Szpiro’s conjecture and Vojta’s conjecture.

Some consequences

The abc conjecture has a large number of consequences. These include both known results, and conjectures for which it gives a conditional proof.

* Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers
* Fermat’s Last Theorem for all sufficiently large exponents (proven in general by Andrew Wiles)
* The Mordell conjecture (Elkies 1991)
* The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)
* The existence of infinitely many non-Wieferich primes (Silverman 1988)
* The weak form of Marshall Hall’s conjecture on the separation between squares and cubes of integers (Nitaj 1996)
* The Fermat–Catalan conjecture, a generalization of Fermat’s last theorem concerning powers that are sums of powers (Pomerance 2008)
* The L function L(s,(−d/.)) formed with the Legendre symbol, has no Siegel zero (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers)
* P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.
* A generalization of Tijdeman’s theorem
* It is equivalent to the Granville–Langevin conjecture.
* It is equivalent to the modified Szpiro conjecture, which would yield a bound of operatorname{rad}(abc)^{frac{6}{5}+epsilon} (Oesterlé 1988).
* Dąbrowski (1996) has shown that the abc conjecture implies that n! + A= k2 has only finitely many solutions for any given integer A.[clarification needed]

While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory.

Abstract. The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichm¨uller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the log-theta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F, together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. In the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves. Finally, we examine the foundational/set-theoretic issues surrounding the vertical and horizontal arrows of the log-theta-lattice by introducing and studying the basic properties of the notion of a “species”, which may be thought of as a sort of formalization, via set-theoretic formulas, of the intuitive notion of a “type of mathematical object”. These foundational issues are closely related to the central role played in the present series of papers by various results from absolute anabelian geometry, as well as to the idea of gluing together distinct models of conventional scheme theory, i.e., in a fashion that lies outside the framework of conventional scheme theory. Moreover, it is precisely these foundational issues surrounding the vertical and horizontal arrows of the log-theta-lattice that led naturally to the introduction of the term “interuniversal”.