Mochizuki abc conjecture error. The following is a consequence of inter-universal Teichmuller theory rem 0. Posted online in 2012, Mochizuki’s papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory. Thus, when self-proclaimed “inter-universal geometer” Mochizuki posted his purported proof of the statement, experts in the field were eager to read through it, despite it being over 500 pages in length. Even Wiles' celebrated 1995 proof of Fermat's Last theorem was flawed when he first publicized it. [1][2] It is stated in terms of three positive integers and (hence the name) that are relatively prime and satisfy . Write on X. He is an expert in arithmetic geometry, a subfield of number theory which provides geometric formulations of the ABC Conjecture (the viewpoint studied in Mochizuki’s work). Eight years ago, Mochizuki posted four massive papers online, claiming to have solved the abc conjecture. We present a fully deterministic operator-theoretic framework establishing the ABC Conjecture. , d A German mathematician has pointed out Mochizuki failed to provide a sufficient proof of the ABC Conjecture. 16 of [2]), the proof is executed on the Fermat curves Cn : {Xn + Y n = Zn} ⊂ P2 Mar 25, 2024 · To summarize the situation before yesterday, virtually all experts in this subject have long ago given up on the idea that Mochizuki’s IUT theory has any hope of proving the abc conjecture. 4. In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a proof of a long-standing problem called the ABC conjecture. 10] and its proof is all of step (v). It would be the biggest flex in mathematical history, maybe all of modern academic history. This is the last equation on [Mochizuki, 2021d, Step (v) on Page 658, Proof of Theorem 1. There was just one problem: His proof, which was more than 500 pages long, was completely impenetrable. My commentary on the geometric case (cited above) demonstrates how closely the proof in the geometric case parallels what M Kirti Joshi, Algebraization of Mochizuki’s anabelian variation of ring structures, perfectoid geometry and formal groups (arxiv:1906. 01228) A 2007 proof of the abc-conjecture by Szpiro turned out to be wrong. A reminder: in terms of the abc -triple, Δ is essentially (abc) 2, and N = rad (abc)). The work baffled mathematicians, who spent years trying to understand it. Others in the field took notice because Mochizuki is well-known and respected in the math field—few believed he would post a proof of the conjecture if he did not believe he had solved it. We give a survey of S. Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. Brian Conrad is a math professor at Stanford and was one of the participants at the Oxford workshop on Mochizuki’s work on the ABC Conjecture. But the team’s presentation of a new application of IUT Theory may help encourage The most striking claimed application of the theory is to provide a proof for various outstanding conjectures in number theory, in particular the abc conjecture. Mochizuki IUTchI, IUTchII, IUTchIII, IUTchIV, Corollary 2. In 2018, two respected math-ematicians said they were confident they had found a flaw in Mochizuki’s proof — something many saw as a death blow to his claims. Given both the simplicity of statement and the difficulty in proving it, the conjecture has been of particular interest to number theorists. 14 and the (d) ⇒ (a) implication in Theorem 14. Inter-universal Teichmüller theory (IUT) was created by Shinichi Mochizuki at Kyoto University, Japan, in a bid to solve a long-standing problem called the ABC conjecture, which focuses on the INTER-UNIVERSAL TEICHM ̈ULLER THEORY I: CONSTRUCTION OF HODGE THEATERS Shinichi Mochizuki May 2020 The present paper is the first in a series of four papers, the Abstract. Proof. PDF Comments NEW !! (2019-06-28) In 2018, a document Why abc is still a conjecture was written by Peter Scholze and Jakob Stix raising objections to the argument. For instance, a proof of the abc conjecture would improve on a landmark result in number theory. This is very odd. 10 in IUTT IV, thus invalidating Mochizuki's original proof of the abc conjecture? This note outlines a constructive proof of a proposition in Mochizuki's paper "Arithmetic elliptic curves in general position," making a direct use of computable non-critical Belyi maps to effectively reduce the full abc -conjecture to a restricted form. More accurate critiques have appeared as a part of the works of Kirti Joshi on the theory of Arithmetic Teichmuller Spaces which includes, in dimension one and genus one, a precise version of Mochizuki’s IUTT. If Shinichi Mochizuki's 500-page proof stands up to scrutiny, mathematicians say it will After an eight-year struggle, embattled Japanese mathematician Shinichi Mochizuki has finally received some validation. e. Suppose that UX is a hyperbolic curve, i. Until Mochizuki released his work, little progress had been made towards proving the abc conjecture since it was proposed in 1985. Mochizuki’s approach to the abc conjecture translates the problem into a question about elliptic curves, a special type of cubic equation in two variables, x and y. Mochizuki isn't asking us to trust that abc is true, even though it may seem that way: he's laid out the mathematics for all to see and check, and, irrespective of its correctness or the amount of time you supposedly have to put in to understand it, it isn't lies. The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. Mochizuki and Prof. Spanning a million lines of Lean, the proof validated Shinichi Mochizuki's Inter-universal Teichmüller theory, with key bridging insights finally making it accessible to the broader mathematical community. Mochizuki announced a proof of the ABC conjecture in 2012 that was finally published in 2021, but as of this writing, most number In March 2018, the authors spent a week in Kyoto at RIMS of intense and constructive discussions with Prof. ” After an eight-year struggle, embattled Japanese mathematician Shinichi Mochizuki has finally received some validation. However, mathematicians understood early on that the conjecture was intertwined with other big problems in mathematics. A canonical Riemann-Hypothesis-inspired Hamiltonian HABC is constructed as a strongresolvent limit, with primes deterministically propagated via the Standing/Sitting Band (SSB)framework. 12, seen as a vital part of Mochizuki’s efforts to solve the abc conjecture, which Scholze and Stix claimed suffered from an unjustified leap of logic. Charles Hoskinson on What are your thoughts on Shinichi Mochizuki purported proof of ABC conjectureFor full Video watch on Charles channel https://youtu. In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a proof of a long standing problem Mochizuki worked alone for 20 years to put together the IUT Theory, but it took more than seven years before an international journal published his article in spring 2021 as the proof for the ABC Does the mistake in Proposition 6. Following an amplification idea of Vojta for deducing the strong abc-conjecture from his own conjecture with ramification for curves (see 14. Question 26. Six years later the jury is still out on whether it’s correct. PDF [3] The Local Pro-p Anabelian Geometry of Curves. Trace-class positivity and spectral rigidity enforce the ABC inequalityc < Kϵ rad (abc)1+ϵ, ∀ coprime triples a There is a fascinating story that I'm sure a lot of you have followed. PDF Comments NEW !! (2012-12-20) [2] A Version of the Grothendieck Conjecture for p-adic Local Fields. Because we have agreed-upon foundations, that sort of thing can't really happen in modern math. 01890) Kirti Joshi, The Absolute Grothendieck Conjecture is false for Fargues-Fontaine Curves (arxiv:2008. Jun 5, 2025 · That’s pretty cool-sounding, but what made this a particularly big deal was that Mochizuki was claiming to have used IUT to solve the abc conjecture, a deep problem in number theory that is We give a survey of S. 10. His 600-page proof of the abc conjecture, one of the biggest open problems If Mochizuki is confident that he is right, he can not only vindicate his credit for solving ABC, but publicly “own” (pwn!) the entire mathematics community simply by answering questions in a public, archived forum. I believe that more voices should participate in the discussion of the abc-conjecture because the public comments by both Mochizuki and Scholze (and some others who have echoed Scholze) make it clear that they simply do not wish to be second-guessed on this matter. Doesn’t the main theorem of [Mochizuki, 2015, Theorem 1. be/w > The error concerned a part of the proof called Conjecture 3. A similar statement can be made concerning S. Jun 13, 2025 · Shinichi Mochizuki’s controversial mathematical proof of abc conjecture divides math experts amid cultural conflicts and competing claims. Note that this is a very different story than the Mochizuki/abc conjecture story: Zhang’s argument use conventional methods and is written out carefully in a manner that should allow experts to readily follow it and check it. Mochizuki’s proof of the abc Con-jecture. . The exposition was designed to be as self-contained as possible. 06840) Kirti Joshi, On Mochizuki’s idea of Anabelomorphy and its applications (arxiv:2003. And anyway, the objections to Mochizuki's work are not philosophical. A side remark: note that the inverse 1 / ℓ of the prime level from the de Rham-Etale correspondence (E †, <ℓ) ↔ E [ℓ] in Mochizuki’s “Hodge-Arakelov theory” ultimately figures as the ϵ in the ABC conjecture. Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community. Mochizuki's ingenious inter-universal Teichmuller theory and ex-plain how it gives rise to Diophantine inequalities. 7 also invalidate Mochizuki's original proof of Theorem 1. 9] contradict [Mochizuki, 2021b] because that theorem asserts that a smooth, hyperbolic p-adic curve over a number field is determined uniquely (up to isomorphism) by its tempered fundamental group? In this section I will give a case study one of the most fascinating current episodes in mathematics, that of Shinichi Mochizuki and his controversial proof of theabc conjecture. 10 in IUTT IV, thus invalidating Mochizuki's original proof of the abc conjecture? Is the ABC Conjecture finally proven? It is a mathematical epic five years in the making . I'm surprised some people think this is a matter of checking whether formal sentences match some English sentences lol It is a matter of checking whether formal sentences match mathematical statements, which are written in natural language. This note outlines a constructive proof of a proposition in Mochizuki's paper Arithmetic elliptic curves in general position, making a direct use of The ABC conjecture Shinichi Mochizuki of the Research Institute for Mathematical Sciences at Kyoto University is such a mathematician. However, my question has nothing to do with the correctness o There still are not many (any?) arithmetic geometers who claim to understand Mochizuki's proof and who still actively engage with the rest of the arithmetic geometry community. His 600-page proof of the abc conjecture, one of the biggest open problems In 2012, the mathematician Shinichi Mochizuki claimed he had solved the abc conjecture, a major open question in number theory about the relationship between addition and multiplication. Hoshi about the suggested proof of the abc conjecture. In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a massive proof claiming to have solved a long-standing problem called the ABC conjecture. It went on to make its seminal contribution in early 2026 with a complete, verifiable proof of the abc conjecture. Kirti Joshi is attempting to reconstruct Mochizuki's proof using Arithmetic Teichmüller Theory, which incorporates perfectoid spaces. I'm aware you're not quite saying this but you seem really distracted Anabelian Geometry, the Geometry of Categories [1] The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves over Number Fields. 1. In 2012, a top mathematician, Shinichi Mochizuki[1], has claimed to have solved the ABC conjecture[2] (an important longstanding problem in number theory), using his own very unique, complex, and abstract Inter-universal Until Mochizuki released his work, little progress had been made towards proving the abc conjecture since it was proposed in 1985. Since he was asked… But, while “I’m not saying Mochizuki will never prove the ABC Conjecture,” she declared, “he hasn’t yet. 3 Let X be a proper smooth geometrically connected curve over a number eld, D X a reduced divisor, UX X D. Does the mistake in Proposition 6. Vojtas conjecture Voj for curves, proved by S. In August 2012, he posted a series of four papers on his personal web page claiming to prove the ABC conjecture, an important outstanding problem in number theory. The discussion centers on the abc conjecture and its connection to Virasoro algebra through Shinichi Mochizuki's Inter-Universal Teichmüller Theory. A Japanese mathematician claims to have the proof for the ABC conjecture, a statement about the relationship between prime numbers that has been called the most important unsolved problem in number theory. Indeed, once the reader admits the main results of the preparatory papers (especially [AbsTopIII], [EtTh]), the numerous constructions in the series of papers [IUTchI], [IUTchII], [IUTchIII], [IUTchIV] on inter-universal Teichm ̈uller theory are likely to is the proof of Mordell’s Conjecture. This note outlines a constructive proof of a proposition in Mochizuki's paper "Arithmetic elliptic curves in general position," making a direct use of computable non-critical Belyi maps to effectively reduce the full abc -conjecture to a restricted form. There is a specific objection to the proof, raised by multiple independent people, that Mochizuki has not been able to respond to adequately. This is equivalent to a modified version of Szpiro’s conjecture in which one replaces min(E) with max(jAj3; B2), where A and B are the coefficients in a short Weierstrass equation for E : y2 = x3 + Ax + B. Another weird feature is that nobody has yet found a way to extend Mochizuki's results, or to apply them to anything other than this specific goal. Davide Castelvecchi at Nature has the story this morning of a press conference held earlier today at Kyoto University to announce the publication by Publications of the Research Institute for Mathematical Sciences (RIMS) of Mochizuki’s purported proof of the abc conjecture. Imagine someone saying "just write good English lol eventually you can do good math". iyna, xfj8ew, p28jy, vi4l1, 8hjfe, hjcpuz, ozmq, 9y4km, skvsv, eofc,