“The abc conjecture in number theory is true.”

Presently, the conjecture is far from being proved; not a single $\epsilon $ is known for which (1.1) holds. … A proof of the $abc$ conjecture is claimed by S. Mochizuki, but this has not been accepted by the general mathematical community [Reference Scholze and Stix8].

The celebrated abc conjecture of Masser and Oesterlé asserts that for any ε > 0 there is a constant K ε > 0 such that every triple (a, b, c) ∈ N 3 of coprime integers solving the equation a + b = c must also satisfy rad(abc) > K ε c 1-ε .

Unfortunately, the argument given for Corollary 3.12 is not a proof, and the theory built in these papers is clearly insufficient to prove the ABC conjecture.

We, the authors of this note, came to the conclusion that there is no proof. We are going to explain where, in our opinion, the suggested proof has a problem, a problem so severe that in our opinion small modifications will not rescue the proof strategy.

And although Mochizuki has an excellent reputation, consensus in the field suggests his arguments do not prove the abc conjecture, meaning the puzzle of a + b = c remains open.* … It therefore remains to be seen whether Kawakami’s efforts will ever bear fruit and the abc conjecture will be settled.

Posted online in 2012, Mochizuki’s papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory. … “I think the abc conjecture is still open,” Scholze said. “Anybody has a chance of proving it.” … The pair “came to the conclusion that there is no proof,” they wrote in their report.

Mathematicians have been working on using a computer to check an apparent proof of the ABC conjecture, but an interim report says there are still serious barriers to overcome. … As a result, there is a small group of people who vehemently argue the proof is sound, but others remain unconvinced. … “Most people believe that there is a serious gap,” says Saha. “And I think that this particular report is fully consistent with that: it has not managed to formalise it, which is what we would expect if this big theory had some serious gaps.”

While a small number of mathematicians have since accepted that Mochizuki’s papers prove the conjecture, other researchers say there are holes in his argument and it needs further work, dividing the mathematical community in two and prompting a prize of up to $1 million for a resolution to the quandary.

The existence of these threefolds implies that the Weakly Special Conjecture formulated in 2000 contradicts the Orbifold Mordell Conjecture, and hence the abc conjecture.

Just a reminder that the abc conjecture is still a conjecture, there is no known valid proof (don’t believe what you might read in an EMS journal).

at least with regard to the substantive mathematical aspects of such a verification, the verification of IUTeich is, for all practical purposes, com plete; nevertheless, as a precautionary measure, in light of the importance of the theory and the novelty of the techniques that underlie the theory, it seems appropriate that a bit more time be allowed to elapse before a final official declaration of the completion of the verification of IUTeich is made.

For over a decade, mathematicians have failed to agree whether a 500-page proof is actually correct. … Then in 2018, two prominent German mathematicians, Peter Scholze at the University of Bonn and Jakob Stix at Goethe University Frankfurt, announced they had located a possible chink in the proof’s armour. … But Mochizuki rejected their argument and, with no grand adjudicating body to rule on who was right or wrong, the validity of IUT theory froze into two camps: on one side, most of the mathematical community; on the other, a small group of researchers loosely affiliated with Mochizuki

Mochizuki’s remarkable claim of the proof of the $abc$ -conjecture rests on an astounding assertion that there exists a Teichmüller Theory222Teichmüller Theory should not be confused with Riemann’s Moduli Theory of Riemann surfaces. of number fields (that is why the phrase Teichmüller Theory appears in the title of [Mochizuki, 2021a, b, c, d]). Mochizuki correctly surmised that such a theory exists, however, that is not enough to prove the existence of such a theory nor are Mochizuki’s anabelian geometry methods adequate for demonstrating its existence.

It’s completely unheard of for a major journal to publish a proof of an important result when experts have publicly stated that the proof is flawed and are standing behind that statement. … I asked around this morning and no one I know who is well-informed about this has heard of any reason to change their opinion that Mochizuki does not have a proof.

According to Nature News, 10 September 2012, quoting Dorian Goldfeld, the abc Conjecture is “the most important unsolved problem in Diophantine analysis”.

The abc conjecture asserts that, for any ε > 0, such triples satisfy rad(abc) ⩾ c 1− ε with finitely many exceptions. … The well-known abc conjecture of Masser and Oesterl´e asserts that, for any λ < 1, there are only finitely many abc triples of exponent λ.

Various attempts to prove the abc conjecture have been made, but none have gained broad acceptance. … Shinichi Mochizuki claimed to have a proof in 2012, but the conjecture is still regarded as unproven by the mainstream mathematical community.

Next, we state the abc Conjecture. Theorem 0.2 (abc Conjecture). For γ ∈ R>0, there exists a Cγ ∈ R>0 such that, for every abc-triple (a, b, c), the following inequality holds: max{| a| , | b| , | c|} < CγN 1+γ (a,b,c).

Due to its profound implications, this simple-to-state conjecture is one of the most important open questions in number theory.

The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable.

Mochizuki and a cadre of close colleagues, mostly at Kyoto University, maintained the proof was correct, while a larger part of the mathematical community argued that the proof was at best indecipherable and at worst fatally flawed.

It is claimed (IUTT IV) that a proof of the abc conjecture can be given in IUTT. In 2018, a document Why abc is still a conjecture was written by Peter Scholze and Jakob Stix raising objections to the argument.

At this stage, we have not been able to conclude whether the abc conjecture has been proven by IUT theory.

The abc conjecture is a conjecture due to Oesterlé and Masser in 1985. … If this conjecture were true, it would imply Fermat's last theorem for sufficiently large powers(Goldfeld 1996).

What is the ABC conjecture? … The ABC conjecture: For each > 0, there are at most finitely many coprime triples a, b, c of positive integers with a + b = c and rad(abc) < c1− . … I like to joke that it has been shown that everything can be proved starting from a false statement, so perhaps the ABC conjecture is more and more resembling one!

A conjectural relationship between the prime factors of two integers and those of their sum, proposed by David Masser and Joseph Oesterlé in 1985.

Shinichi Mochizuki anounced the proof which the mathematical community perceives as a serious but unchecked claim. … Kirti Joshi has put out an alleged proof of the abc conjecture on the arXiv: … According to Peter Scholze, there is a mistake in Joshi’s proof at Proposition 6.10.7:

Presently, the conjecture is far from being proved; not a single is known for which (1) holds.

He and Jakob Stix, an expert in the relevant field of anabelian geometry, visited Mochizuki in March 2018. In the end, they both remained convinced that there is an unfillable gap in the proof, while Mochizuki keeps claiming that it is true. … The persistent confusion over the status of the proof remains even in 2023, showing no sign of abating with one part of the mathematical community trying to build additional work over the method used and another part denying any value to the proof.

We study the size of the exceptional set in the $abc$ conjecture, improving on and simplifying work of Browning, Lichtman and Teräväinen.

ABC Conjecture (Masser (1985), Oesterlé (1988)) Suppose ą 0. Then there are finitely many abc-triples with quality greater than 1 ` .

The abc Conjecture Conjecture (David Masser and Joseph Oesterl´e, 1985.)

The abc conjecture says that if a, b, c is a large abc triple then abc cannot be ‘very composite’.

The abc Conjecture Conjecture (David Masser and Joseph Oesterl´e, 1985.)