Английская Википедия:Abc conjecture

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Шаблон:Short description Шаблон:Infobox mathematical statement

Файл:Oesterle Joseph.jpg
Mathematician Joseph Oesterlé
Файл:David Masser.jpg
Mathematician David Masser

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.Шаблон:SfnШаблон:Sfn It is stated in terms of three positive integers <math>a, b</math> and <math>c</math> (hence the name) that are relatively prime and satisfy <math>a+b=c</math>. The conjecture essentially states that the product of the distinct prime factors of <math>abc</math> is usually not much smaller than <math>c</math>. 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".Шаблон:Sfn

The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves,[1] 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.Шаблон:Sfn

Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community, and, as of 2023, the conjecture is still regarded as unproven.[2][3]

Formulations

Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integer <math>n</math>, the radical of <math>n</math>, denoted <math>\text{rad}(n)</math>, is the product of the distinct prime factors of <math>n</math>. For example,

<math>\text{rad}(16)=\text{rad}(2^4)=\text{rad}(2)=2</math>

<math>\text{rad}(17)=17</math>

<math>\text{rad}(18)=\text{rad}(2\cdot 3^2)=2\cdot3 =6</math>

<math>\text{rad}(1000000)=\text{rad}(2^6 \cdot 5^6)=2\cdot5=10</math>

If a, b, and c are coprime[notes 1] positive integers such that a + b = c, it turns out that "usually" <math>c<\text{rad}(abc)</math>. The abc conjecture deals with the exceptions. Specifically, it states that:

Шаблон:Block indent

An equivalent formulation is:

Шаблон:Block indent

Equivalently (using the little o notation):

Шаблон:Block indent

A fourth equivalent formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), which is defined as

Шаблон:Block indent

For example:

Шаблон:Block indent

A typical triple (a, b, c) of coprime positive integers with a + b = c will have c < rad(abc), i.e. q(a, b, c) < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. The fourth formulation is:

Шаблон:Block indent

Whereas it is known that there are infinitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or even q > 1.0001, etc. In particular, if the conjecture is true, then there must exist a triple (a, b, c) that achieves the maximal possible quality q(a, b, c).

Examples of triples with small radical

The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with c > rad(abc). For example, let

Шаблон:Block indent

The integer b is divisible by 9:

Шаблон:Block indent

Using this fact, the following calculation is made:

Шаблон:Block indent

By replacing the exponent 6n with other exponents forcing b to have larger square factors, the ratio between the radical and c can be made arbitrarily small. Specifically, let p > 2 be a prime and consider

Шаблон:Block indent

Now it may be plausibly claimed that b is divisible by p2:

Шаблон:Block indent

The last step uses the fact that p2 divides 2p(p−1) − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2p(p−1) = p2(...) + 1.

And now with a similar calculation as above, the following results:

Шаблон:Block indent

A list of the highest-quality triples (triples with a particularly small radical relative to c) is given below; the highest quality, 1.6299, was found by Eric Reyssat Шаблон:Harv for Шаблон:Block indent Шаблон:Block indent Шаблон:Block indent Шаблон:Block indent

Some consequences

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:

Theoretical results

The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. Bounds are known that are exponential. Specifically, the following bounds have been proven:

Шаблон:Block indent Шаблон:Block indent Шаблон:Block indent\left(\log(\operatorname{rad}(abc)\right)^3\right) } </math> Шаблон:Harv.}}

In these bounds, K1 and K3 are constants that do not depend on a, b, or c, and K2 is a constant that depends on ε (in an effectively computable way) but not on a, b, or c. The bounds apply to any triple for which c > 2.

There are also theoretical results that provide a lower bound on the best possible form of the abc conjecture. In particular, Шаблон:Harvtxt showed that there are infinitely many triples (a, b, c) of coprime integers with a + b = c and

Шаблон:Block indent

for all k < 4. The constant k was improved to k = 6.068 by Шаблон:Harvtxt.

Computational results

In 2006, the Mathematics Department of Leiden University in the Netherlands, together with the Dutch Kennislink science institute, launched the ABC@Home project, a grid computing system, which aims to discover additional triples a, b, c with rad(abc) < c. Although no finite set of examples or counterexamples can resolve the abc conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.

Distribution of triples with q > 1[9]
scope="col" Шаблон:Diagonal split header q > 1 q > 1.05 q > 1.1 q > 1.2 q > 1.3 q > 1.4
c < 102 6 4 4 2 0 0
c < 103 31 17 14 8 3 1
c < 104 120 74 50 22 8 3
c < 105 418 240 152 51 13 6
c < 106 1,268 667 379 102 29 11
c < 107 3,499 1,669 856 210 60 17
c < 108 8,987 3,869 1,801 384 98 25
c < 109 22,316 8,742 3,693 706 144 34
c < 1010 51,677 18,233 7,035 1,159 218 51
c < 1011 116,978 37,612 13,266 1,947 327 64
c < 1012 252,856 73,714 23,773 3,028 455 74
c < 1013 528,275 139,762 41,438 4,519 599 84
c < 1014 1,075,319 258,168 70,047 6,665 769 98
c < 1015 2,131,671 463,446 115,041 9,497 998 112
c < 1016 4,119,410 812,499 184,727 13,118 1,232 126
c < 1017 7,801,334 1,396,909 290,965 17,890 1,530 143
c < 1018 14,482,065 2,352,105 449,194 24,013 1,843 160

As of May 2014, ABC@Home had found 23.8 million triples.[10]

Шаблон:Visible anchor[11]
Rank q a b c Discovered by
1 1.6299 2 310·109 235 Eric Reyssat
2 1.6260 112 32·56·73 221·23 Benne de Weger
3 1.6235 19·1307 7·292·318 28·322·54 Jerzy Browkin, Juliusz Brzezinski
4 1.5808 283 511·132 28·38·173 Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
5 1.5679 1 2·37 54·7 Benne de Weger

Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.

Refined forms, generalizations and related statements

The abc conjecture is an integer analogue of the Mason–Stothers theorem for polynomials.

A strengthening, proposed by Шаблон:Harvtxt, states that in the abc conjecture one can replace rad(abc) by

Шаблон:Block indent

where ω is the total number of distinct primes dividing a, b and c.Шаблон:Sfnp

Andrew Granville noticed that the minimum of the function <math>\big(\varepsilon^{-\omega}\operatorname{rad}(abc)\big)^{1+\varepsilon}</math> over <math>\varepsilon > 0</math> occurs when <math>\varepsilon = \frac{\omega}{\log\big(\operatorname{rad}(abc)\big)}.</math>

This inspired Шаблон:Harvtxt to propose a sharper form of the abc conjecture, namely: Шаблон:Block indent with κ an absolute constant. After some computational experiments he found that a value of <math>6/5</math> was admissible for κ. This version is called the "explicit abc conjecture".

Шаблон:Harvtxt also describes related conjectures of Andrew Granville that would give upper bounds on c of the form

Шаблон:Block indent

where Ω(n) is the total number of prime factors of n, and

Шаблон:Block indent

where Θ(n) is the number of integers up to n divisible only by primes dividing n.

Шаблон:Harvtxt proposed a more precise inequality based on Шаблон:Harvtxt. Let k = rad(abc). They conjectured there is a constant C1 such that

Шаблон:Block indent\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{1}}{\log\log k}\right)\right)</math>}}

holds whereas there is a constant C2 such that

Шаблон:Block indent\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{2}}{\log\log k}\right)\right)</math>}}

holds infinitely often.

Шаблон:Harvtxt formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.

Claimed proofs

Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards.[12]

Since August 2012, Shinichi Mochizuki has claimed a proof of Szpiro's conjecture and therefore the abc conjecture.[13] 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.[14] The papers have not been widely accepted by the mathematical community as providing a proof of abc.[15] This is not only because of their length and the difficulty of understanding them,[16] but also because at least one specific point in the argument has been identified as a gap by some other experts.[17] Although a few mathematicians have vouched for the correctness of the proof[18] and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.[19][20]

In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki.[21][22] 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";[17] Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.[23][24][25]

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.[2] 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".[2] In March 2021, Mochizuki's proof was published in RIMS.[26]

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.[27]

See also

Notes

Шаблон:Reflist

References

Шаблон:Reflist

Sources

Шаблон:Refbegin

Шаблон:Refend

External links

  1. Шаблон:Cite journal
  2. 2,0 2,1 2,2 Шаблон:Cite journal
  3. Further comment by P. Scholze at Not Even Wrong math.columbia.eduШаблон:Self-published inline
  4. Шаблон:Cite journal
  5. The ABC-conjecture, Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.
  6. Шаблон:Harvtxt; Шаблон:Harvtxt
  7. Шаблон:Citation
  8. Шаблон:ArXiv Andrea Surroca, Siegel’s theorem and the abc conjecture, Riv. Mat. Univ. Parma (7) 3, 2004, S. 323–332
  9. Шаблон:Citation.
  10. Шаблон:Citation
  11. Шаблон:Cite web
  12. "Finiteness Theorems for Dynamical Systems", Lucien Szpiro, talk at Conference on L-functions and Automorphic Forms (on the occasion of Dorian Goldfeld's 60th Birthday), Columbia University, May 2007. See Шаблон:Citation.
  13. Шаблон:Cite journal
  14. Шаблон:Cite journal
  15. Шаблон:Cite web
  16. Шаблон:Cite magazine
  17. 17,0 17,1 Шаблон:Cite web (updated version of their May report Шаблон:Webarchive)
  18. Шаблон:Cite journal
  19. Шаблон:Cite web
  20. Шаблон:Cite journal
  21. Шаблон:Cite magazine
  22. Шаблон:Cite web Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material
  23. Шаблон:Cite web
  24. Шаблон:Cite web
  25. Шаблон:Cite web
  26. Шаблон:Cite web
  27. Шаблон:Cite web


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