Katana VentraIP

abc conjecture

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.[1][2] It is stated in terms of three positive integers and (hence the name) that are relatively prime and satisfy . The conjecture essentially states that the product of the distinct prime factors of is usually not much smaller than . A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".[3]

Katana VentraIP

$_$_$DEEZ_NUTS#0__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#0__subtitleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#0__call_to_action.textDEEZ_NUTS$_$_$

The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves,[4] which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture.[1]


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.[5][6][7]

Claimed proofs[edit]

Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards.[28]


Since August 2012, Shinichi Mochizuki has claimed a proof of Szpiro's conjecture and therefore the abc conjecture.[5] He released a series of four preprints developing a new theory he called inter-universal Teichmüller theory (IUTT), which is then applied to prove the abc conjecture.[29] The papers have not been widely accepted by the mathematical community as providing a proof of abc.[30] This is not only because of their length and the difficulty of understanding them,[31] but also because at least one specific point in the argument has been identified as a gap by some other experts.[32] Although a few mathematicians have vouched for the correctness of the proof[33] and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.[34][35]


In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki.[36][37] While they did not resolve the differences, they brought them into clearer focus. Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that ... small modifications will not rescue the proof strategy";[32] Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.[38][39][40]


On April 3, 2020, two mathematicians from the Kyoto research institute where Mochizuki works announced that his claimed proof would be published in Publications of the Research Institute for Mathematical Sciences, the institute's journal. Mochizuki is chief editor of the journal but recused himself from the review of the paper.[6] The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp".[6] In March 2021, Mochizuki's proof was published in RIMS.[41]

$_$_$DEEZ_NUTS#7__descriptionDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#2__descriptionDEEZ_NUTS$_$_$

The (already proven in general by Gerd Faltings).[10]

Mordell conjecture

As equivalent, in dimension 1.[11]

Vojta's conjecture

The allowing for a finite number of counterexamples.[12]

Erdős–Woods conjecture

The existence of infinitely many non- in every base b > 1.[13]

Wieferich primes

The weak form of on the separation between squares and cubes of integers.[14]

Marshall Hall's conjecture

has a famously difficult proof by Andrew Wiles. However it follows easily, at least for , from an effective form of a weak version of the abc conjecture. The abc conjecture says the lim sup of the set of all qualities (defined above) is 1, which implies the much weaker assertion that there is a finite upper bound for qualities. The conjecture that 2 is such an upper bound suffices for a very short proof of Fermat's Last Theorem for .[15]

Fermat's Last Theorem

The , a generalization of Fermat's Last Theorem concerning powers that are sums of powers.[16]

Fermat–Catalan conjecture

The L(s, χd) formed with the Legendre symbol, has no Siegel zero, given a uniform version of the abc conjecture in number fields, not just the abc conjecture as formulated above for rational integers.[17]

L-function

A P(x) has only finitely many perfect powers for all integers x if P has at least three simple zeros.[18]

polynomial

A generalization of concerning the number of solutions of ym = xn + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Aym = Bxn + k.

Tijdeman's theorem

As equivalent, the Granville–Langevin conjecture, that if f is a square-free binary form of degree n > 2, then for every real β > 2 there is a constant C(f, β) such that for all coprime integers x, y, the radical of f(x, y) exceeds C · max{|x|, |y|}nβ.

[19]

all the polynominals (x^n-1)/(x-1) have an infinity of square-free values.

[20]

The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a conditional proof. The consequences include:

$_$_$DEEZ_NUTS#5__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#5__descriptionDEEZ_NUTS$_$_$

List of unsolved problems in mathematics

Distributed computing project called ABC@Home.

ABC@home

: Easy to follow, detailed explanation by Brian Hayes.

Easy as ABC

"abc Conjecture". MathWorld.

Weisstein, Eric W.

Abderrahmane Nitaj's

ABC conjecture home page

Bart de Smit's

ABC Triples webpage

http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf

by Noam D. Elkies

The ABC's of Number Theory

by Barry Mazur

Questions about Number

on MathOverflow

Philosophy behind Mochizuki’s work on the ABC conjecture

Polymath project wiki page linking to various sources of commentary on Mochizuki's papers.

ABC Conjecture

Numberphile video

abc Conjecture

News about IUT by Mochizuki

$_$_$DEEZ_NUTS#3__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__quote--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__name--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#3__company_or_position--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--7DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--8DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--9DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--9DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--9DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--10DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--10DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--10DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--11DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--11DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--11DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--12DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--12DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--12DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__quote--13DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__name--13DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#6__company_or_position--13DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#9__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#4__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__quote--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__name--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#8__company_or_position--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__titleDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__subtextDEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--0DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--1DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--2DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--3DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--4DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--5DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--6DEEZ_NUTS$_$_$

$_$_$DEEZ_NUTS#1__answer--7DEEZ_NUTS$_$_$