abc conjecture

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French mathematician Joseph Oesterlé
British mathematician David Masser

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is usually not much smaller than c. In other words: if a and b are composed from large powers of primes, then c is usually not divisible by large powers of primes. The precise statement is given below.

The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves.[1] The latter conjecture has more geometric structures involved in its statement in comparison with the abc conjecture

The abc conjecture and its versions express, in concentrate form, some fundamental feature of various problems in Diophantine geometry. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".

Lucien Szpiro attempted a solution in 2007, but it was found not to be correct.[2]

In August 2012 Shinichi Mochizuki posted his four preprints which develop a entirely new inter-universal Teichmüller theory (IUT). The fourth paper applied IUT to obtain the proof of several famous conjectures including the abc conjecture. Mochizuki's papers were submitted to a mathematical journal and are being refereed. Substantial efforts have been undertaken to help other mathematicians to study this groundbreaking theory. Due to its novelty, the number of experts on it in 2012 was zero. To assist mathematicians to study IUT, two international conferences were organized.[3] In 2017 the number of experts on IUT is between 10 and 20, and no major mistakes have been found.[4]

Formulations[edit]

Before we state the conjecture we need to introduce the notion of the radical of an integer: for a positive integer n, the radical of n, denoted rad(n), is the product of the distinct prime factors of n. For example

rad(16) = rad(24) = 2,
rad(17) = 17,
rad(18) = rad(2 ⋅ 32) = 2 · 3 = 6,
rad(1000000) = rad(26 ⋅ 56) = 2 ⋅ 5 = 10.

If a, b, and c are coprime[5] positive integers such that a + b = c, it turns out that "usually" c < rad(abc). The abc conjecture deals with the exceptions. Specifically, it states that:

ABC Conjecture. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that:

An equivalent formulation states that:

ABC Conjecture II. For every ε > 0, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:

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

For example,

q(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
q(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...

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.

ABC Conjecture III. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.

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) which achieves the maximal possible quality q(a, b, c) .

Examples of triples with small radical[edit]

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

First we note that b is divisible by 9:

Using this fact we calculate:

By replacing the exponent 6n by 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:

Now we claim that b is divisible by p2:

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 we have:

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 (Lando & Zvonkin 2004, p. 137) for

a = 2,
b = 310·109 = 6,436,341,
c = 235 = 6,436,343,
rad(abc) = 15042.

Some consequences[edit]

The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately since the conjecture has been stated) and conjectures for which it gives a conditional proof. While an earlier proof of the conjecture would have been more significant in terms of consequences, the abc conjecture itself remains of interest for the other conjectures it would prove, together with its numerous links with deep questions in number theory.

  • Fermat's Last Theorem 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 .[9]
  • The Beal conjecture, a generalization of Fermat's last theorem proposing that if A, B, C, x, y, and z are positive integers with Ax + By = Cz and x, y, z > 2, then A, B, and C have a common prime factor.

Theoretical results[edit]

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

(Stewart & Tijdeman 1986),
(Stewart & Yu 1991), and
(Stewart & Yu 2001).

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

Computational results[edit]

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[10]
  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

ABC@Home had found 23.8 million triples.[11]

Highest quality triples[12]
  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[edit]

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

A strengthening, proposed by Baker (1998), states that in the abc conjecture one can replace rad(abc) by

εω rad(abc),

where ω is the total number of distinct primes dividing a, b and c (Bombieri & Gubler 2006, p. 404).

Andrew Granville noticed that the minimum of the function over occurs when

This incited Baker (2004) to propose a sharper form of the abc conjecture, namely:

with κ an absolute constant. After some computational experiments he found that a value of was admissible for κ.

This version is called "explicit abc conjecture".

From the previous inequality, Baker deduced a stronger form of the original abc conjecture: let a, b, c be coprime positive integers with a + b = c; then we have:

.

Baker (1998) also describes related conjectures of Andrew Granville that would give upper bounds on c of the form

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

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

Robert, Stewart & Tenenbaum (2014) proposed more precise inequality based on Robert & Tenenbaum (2013). Let k = rad(abc). They conjectured there is a constant C1 such that

holds whereas there is a constant C2 such that

holds infinitely often.

Browkin & Brzeziński (1994) formulated the n conjecture—a version of the abc conjecture involving n > 2 integers.

The work of Shinichi Mochizuki[edit]

In August 2012, Shinichi Mochizuki released a series of four preprints on Inter-universal Teichmuller Theory which is then applied to prove several famous conjectures in number theory, including Szpiro's conjecture, the hyperbolic Vojta's conjecture and the abc conjecture.[13] Mochizuki calls the theory on which this proof is based "inter-universal Teichmüller theory (IUT)". The theory is radically different from any standard theory and goes well outside the scope of arithmetic geometry. It was developed over two decades with the last four IUT papers[14][15][16][17] occupying the space of over 500 pages and using many of his prior published papers.[18]

Mochizuki released progress reports in December 2013[19] and December 2014.[20] He has invested hundreds of hours to run seminars and meetings to discuss his theory.[21] According to Mochizuki, verification of the core proof is "for all practical purposes, complete." However, he also stated that an official declaration should not happen until some time later in the 2010s, due to the importance of the results and new techniques.[20]

The first international workshop on Mochizuki's theory was organized by Ivan Fesenko and held in Oxford in December 2015.[22] It helped to increase the number of mathematicians who had thoroughly studied parts of the IUT papers or related prerequisite papers. The next workshop on IUT Summit was held at the Research Institute for Mathematical Sciences in Kyoto in July 2016.[23] After that workshop at least ten mathematicians now understand the theory in detail.[24] There are several introductory texts and surveys of the theory, written by Mochizuki and other mathematicians.[25]

See also[edit]

Notes[edit]

  1. ^ Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki" (PDF), Europ. J. Math., 1: 405–440 .
  2. ^ "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 Woit, Peter (May 26, 2007), "Proof of the abc Conjecture?", Not Even Wrong .
  3. ^ Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop, July 18-27 2016) 
  4. ^ Revell, Timothy (7 September 2017), "Baffling ABC maths proof now has impenetrable 300-page 'summary'", New Scientist 
  5. ^ When a + b = c, coprimeness of a, b, c implies pairwise coprimeness of a, b, c. So in this case, it does not matter which concept we use.
  6. ^ The ABC-conjecture, Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.
  7. ^ Mollin (2009)
  8. ^ Mollin (2010) p. 297
  9. ^ Granville, Andrew; Tucker, Thomas (2002). "It’s As Easy As abc". Notices of the AMS 49 (10): 1224–1231.
  10. ^ "Synthese resultaten", RekenMeeMetABC.nl (in Dutch), archived from the original on December 22, 2008, retrieved October 3, 2012 .
  11. ^ "Data collected sofar", ABC@Home, archived from the original on May 15, 2014, retrieved April 30, 2014 
  12. ^ "100 unbeaten triples". Reken mee met ABC. 2010-11-07. 
  13. ^ Mochizuki, Shinichi (May 2015). Inter-universal Teichmuller Theory I: Construction of Hodge Theaters, Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation, Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice., Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations, available at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
  14. ^ Mochizuki, Shinichi (2012a), Inter-universal Teichmuller Theory I: Construction of Hodge Theaters (PDF) .
  15. ^ Mochizuki, Shinichi (2012b), Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation (PDF) .
  16. ^ Mochizuki, Shinichi (2012c), Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice (PDF) .
  17. ^ Mochizuki, Shinichi (2012d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF) .
  18. ^ Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki" (PDF), Europ. J. Math., 1: 405–440 .
  19. ^ "On the Verification of Inter-Universal Teichmüller Theory: A Progress Report (as of December 2013)" by Shinichi Mochizuki
  20. ^ a b "On the Verification of Inter-Universal Teichmüller Theory: A Progress Report (as of December 2014)" by Shinichi Mochizuki
  21. ^ Seminars, Meetings, and Lectures on IUT in Japan, School of Mathematical Sciences, University of Nottingham.
  22. ^ "Workshop on IUT Theory of Shinichi Mochizuki, Oxford, December 7-11 2015". School of Mathematical Sciences, University of Nottingham. Retrieved 21 March 2016. 
  23. ^ "IUT Summit, RIMS workshop, July 18-27 2016". School of Mathematical Sciences, University of Nottingham. Retrieved 21 March 2016. 
  24. ^ "Mathematicians finally starting to understand epic ABC proof". New Scientist. Retrieved 6 August 2016. 
  25. ^ "Texts related to IUT". School of Mathematical Sciences, University of Nottingham. Retrieved 21 March 2016. 

References[edit]

External links[edit]