abc conjecture

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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) as an integer analogue of the Mason–Stothers theorem for polynomials. The conjecture is stated in terms of three positive integers, a, b and c (hence the name), which have no common factor 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 has already become well known for the number of interesting consequences it entails. Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".

Several solutions have been proposed to the abc conjecture, the most recent of which is still being evaluated by the mathematical community, though it still remains open as of September 2014.

Formulations[edit]

The abc conjecture can be expressed as follows: For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d1+ε, where d denotes the product of the distinct prime factors of abc.

To illustrate the terms used, if

a = 16 = 24,
b = 17, and
c = 16 + 17 = 33 = 3·11,

then d = 2·17·3·11 = 1122, which is greater than c. Therefore, for all ε > 0, c is not greater than d1+ε. According to the conjecture, most coprime triples where a + b = c are like the ones used in this example, and for only a few exceptions is c > d1+ε.

To add more terminology: 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.

If a, b, and c are coprime[1] 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 positive coprime integers, with a + b = c, such that

c>\operatorname{rad}(abc)^{1+\varepsilon}

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, the inequality

c < K_{\varepsilon} \cdot \operatorname{rad}(abc)^{1+\varepsilon}
holds.

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

 q(a, b, c) = \frac{ \log(c) }{ \log( \operatorname{rad}( abc ) ) }

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.

Examples of triples with small radical[edit]

The condition that ε > 0 is necessary for the truth of the conjecture, as there exist infinitely many triples a, b, c with rad(abc) < c. For instance, such a triple may be taken as

a = 1
b = 26n − 1
c = 26n

As a and c together contribute only a factor of two to the radical, while b is divisible by 9, rad(abc) < 2c/3 for these examples, if n > 1. This is because rad(abc) = rad(a)rad(b)rad(c) =2rad(b). b= 64n - 1n = (64 - 1)(....) = 32 x 7 x (....). So b=32r for some r. So rad(b) = rad(32r) <= 3r = 1/3b. So rad(abc) = 2rad(b) <= 2b/3 < 2c/3.


By replacing the exponent 6n by other exponents forcing b to have larger square factors, the ratio between the radical and c may be made arbitrarily small. Specifically, replacing 6n by p(p − 1)n for an arbitrary prime p > 2 will make b divisible by p2, because 2p(p−1) ≡ 1 (mod p2) and 2p(p−1) − 1 will be a factor of b.

A list of the highest quality triples (triples with a particularly small radical relative to c) is given below; the highest quality of these, with quality 1.6299, was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137):

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[edit]

Fermat's last theorem was proven by Andrew Wiles and the proof is famous for its difficulty. But if a strong effective form of the abc conjecture is correct, the proof of Fermat's Last Theorem becomes much shorter and easier as follows:[5]
If abc conjecture is correct when K=1 and ε=1, and when the co-prime natural numbers A,B,C satisfy an equation A + B = C, we have C < (rad(ABC))2.
We assume the co-prime natural numbers an,bn,cn satisfy an + bn = cn, replacing A to an, B to bn, C to cn. This equation an + bn = cn is the Fermat's Last Theorem. Then we get

c^n < (rad(a^n b^n c^n))^2 = (rad(abc))^2 \le (abc)^2 < (c^3)^2 = c^6.

(rad(x^n)=rad(x),  rad(x) \le x ,  a \cdot b \cdot c < c \cdot c \cdot c \  ).

Now we get

c^n < c^6

That's why n must be smaller than 6. But for exponents n = 3,4,5, we already have proofs which were proved before(Fermat,Euler,Dirichlet or Legendre), so no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two.[6]
In this argument one can allow ε to be smaller and K to be larger, at the cost of requiring explicit checks that there are no small solutions to the Fermat equation. These checks are possible for reasonable values of ε and K, but it is possible (even perhaps likely) that a proof of the abc conjecture will give only ineffective bounds on K in terms of ε. In this case the deduction yields only the weaker statement that there are finitely many solutions to the Fermat equation. This is stronger than Faltings' theorem, which states that there are finitely many solutions to the Fermat equation for each n.

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:

c < \exp{ \left(K_1 \operatorname{rad}(abc)^{15}\right) } (Stewart & Tijdeman 1986),
c < \exp{ \left(K_2 \operatorname{rad}(abc)^{\frac{2}{3} + \varepsilon}\right) } (Stewart & Yu 1991), and
c < \exp{ \left(K_3 \operatorname{rad}(abc)^{\frac{1}{3} + \varepsilon}\right) } (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[7]
  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 April 2014, ABC@Home has found 23.8 million triples, and its present goal is to obtain a complete list of all ABC triples (a,b,c) with c no more than 263.[8]

Highest quality triples[9]
  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 and generalizations[edit]

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

ε−ωrad(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  {\varepsilon}^{-\omega} (\operatorname{rad}(abc))^{1+\varepsilon} over  {\varepsilon}>0 occurs when  {\varepsilon}= \frac{\omega}{\log(\operatorname{rad}(abc))}.

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

c<{\kappa}\operatorname{rad}(abc)\frac{(\log(\operatorname{rad}(abc)))^{\omega}}{\omega!}

with {\kappa} an absolute constant. After some computational experiments in order to find a value for {\kappa}, he found that a value of \frac{6}{5} was admissible.

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 c<(\operatorname{rad}(abc))^{1+\frac{3}{4}}


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

 K^{\Omega(a b c)} \mathrm{rad}(a b c) \

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

 O(\mathrm{rad}(a b c) \Theta(a b c)) \

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

 c<k\exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{1}}{\log\log k}\right)\right)

holds whereas there is a constant C2 such that

 c>k\exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{2}}{\log\log k}\right)\right)

holds infinitely often.

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

Attempts at solution[edit]

Lucien Szpiro attempted a solution in 2007 but it was found to be incorrect.[10]

In August 2012, Shinichi Mochizuki released a series of four preprints containing a claim to a proof of the abc conjecture. Mochizuki calls the theory on which this proof is based "inter-universal Teichmüller theory", and it has other applications including a proof of Szpiro's conjecture and Vojta's conjecture.[11][12] Experts were expected to take months to check Mochizuki's new mathematical machinery, which was developed over decades in 500 pages of preprints and several of his prior papers.[13] Attempts at verifying Mochizuki's work are severely hampered by his refusal to leave his home university and lecture on his new mathematics, as is standard in the academy.[14]

When an error in one of the articles was pointed out by Vesselin Dimitrov and Akshay Venkatesh in October 2012, Mochizuki posted a comment on his website acknowledging the mistake, stating that it would not affect the result, and promising a corrected version in the near future.[15] He revised all of his papers on "inter-universal Teichmüller theory" of which the latest 2 revisions dated December 2013.[11] Mochizuki has refused all requests for media interviews, but released a progress report in December 2013.[16]

See also[edit]

Notes[edit]

  1. ^ Note that when it is given that a + b = c, coprimeness of a, bc implies pairwise coprimeness of abc. So in this case, it does not matter which concept we use.
  2. ^ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf
  3. ^ Mollin (2009)
  4. ^ Mollin (2010) p.297
  5. ^ Granville, Andrew; Tucker, Thomas (2002). “It’s As Easy As abc”. Notices of the AMS 49 (10): 1224–1231.
  6. ^ Yamazaki, Takao"Fermat's Last Theorem and ABC conjecture"(Japanese), Sugaku seminar Oct 2010.
  7. ^ Synthese resultaten, RekenMeeMetABC.nl, retrieved October 3, 2012  (Dutch).
  8. ^ Data collected sofar, ABC@Home, retrieved April 30, 2014 
  9. ^ "100 unbeaten triples". Reken mee met ABC. 2010-11-07. 
  10. ^ "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 .
  11. ^ a b Mochizuki, Shinichi (August 2012). 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
  12. ^ Ball, Phillip (10 September 2012), Proof claimed for deep connection between primes, Nature .
  13. ^ Cipra, Barry (September 12, 2012), ABC Proof Could Be Mathematical Jackpot, Science .
  14. ^ The Paradox of the Proof
  15. ^ Kevin Hartnett (3 November 2012). "An ABC proof too tough even for mathematicians". Boston Globe. 
  16. ^ [1]

References[edit]

External links[edit]