abc conjecture


abc conjecture

The abc conjecture (also known as Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c (whence comes 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 rarely much smaller than c.

Although there is no obvious strategy for resolving the problem, it has already become well known for the number of interesting consequences it entails. Dorian M. Goldfeld described the abc conjecture as "the most important unsolved problem in Diophantine analysis".[1]

Unsolved problems in mathematics

Are there for every ε > 0, only finitely many triples of coprime positive integers

a + b = c

such that

c > d (1+ε)

where d denotes the product of the distinct prime factors of abc?

Contents

Formulations

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[2] 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 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 for any ε > 0, there exists a constant K such that, for all triples of coprime positive integers (a, b, c) satisfying a + b = c, the inequality

c < K \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.

The abc conjecture states that, for any 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

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. 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 large. Another triple with a particularly small radical was found by Eric Reyssat:[3]

a = 2:
b = 310 109 = 6436341
c = 235 = 6436343
rad(abc) = 15042.

Some consequences

The conjecture has not been proven, but it has a large number of interesting consequences. These include both known results, and conjectures for which it gives a conditional proof.

  • Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers
  • Fermat's Last Theorem for all sufficiently large exponents (proven in general by Andrew Wiles)
  • The Mordell conjecture (proven by Gerd Faltings)[4]
  • The Erdős–Woods conjecture except for a finite number of counterexamples[5]
  • The existence of infinitely many non-Wieferich primes[6]
  • The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers[7]
  • The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers[8]
  • The L function L(s,(−d/.)) formed with the Legendre symbol, has no Siegel zero (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers)
  • P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.[9]
  • A generalization of Tijdeman's theorem
  • It is equivalent to the Granville–Langevin conjecture
  • It is equivalent to the modified Szpiro conjecture.
  • Dąbrowski (1996) has shown that the abc conjecture implies that n! + A= k2 has only finitely many solutions for any given integer A.[10]

While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory.

Theoretical results

It remains unknown whether c can be upper bounded by a near-linear function of the radical of abc, as the abc conjecture states, or even whether it can be bounded by a polynomial of rad(abc). However, exponential bounds are known. Specifically, the following bounds have been proven:

c < \exp{(K_1  \operatorname{rad}(abc)^{15}) } (C. L. Stewart & R. Tijdeman 1986),[11]
c < \exp{ (K_2  \operatorname{rad}(abc)^{2/3+\varepsilon}) } (C. L. Stewart & Kunrui Yu 1991),[12] and
c < \exp{ (K_3  \operatorname{rad}(abc)^{1/3+\varepsilon}) } (C. L. Stewart & Kunrui Yu 2001).[13]

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

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[14]
  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 14.482.059 2.352.105 449.194 24.013 1.843 160

As of March 2011, ABC@Home has found 20.9 million triples, and its present goal is to obtain a complete list of all ABC triples (a,b,c) with c no more than 1020.[15]

Highest quality triples[16]
  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 1.5444 72​412​3113 1116​132​79 2·​33​523​953 Abderrahmane Nitaj
5 1.4805 522​79·​45949 32​1318​613 223​174​2512​17333 Frank Rubin

where 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 ) ) }.

Refined forms and generalizations

A stronger inequality proposed in 1996 by Alan Baker 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. A related conjecture of Andrew Granville states that on the RHS we could also put

O(rad(abc) Θ(rad(abc)))

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

In 1994, Jerzy Browkin and Juliusz Brzeziński formulated the n-conjecture[17]—a version of the abc conjecture involving n > 2 integers.

See also

Notes

  1. ^ Goldfeld, Dorian (1996), "Beyond the last theorem", Math Horizons (September): 26–34 
  2. ^ Note that when it is given that a + b = c, coprimeness of abc implies pairwise coprimeness of abc. So in this case, it does not matter which concept we use.
  3. ^ Lando and Zvonkin, p.137
  4. ^ Elkies, N. D. (1991), "ABC implies Mordell", Intern. Math. Research Notices 7: 99–109, doi:10.1155/S1073792891000144 
  5. ^ Langevin, M. (1993), "Cas d'égalité pour le théorème de Mason et applications de la conjecture abc", Comptes rendus de l'Académie des sciences 317 (5): 441–444  (French).
  6. ^ Silverman, Joseph H. (1988), "Wieferich's criterion and the abc-conjecture", Journal of Number Theory 30 (2): 226–237, doi:10.1016/0022-314X(88)90019-4 
  7. ^ Nitaj, Abderrahmane (1996), "La conjecture abc", Enseign. Math. 42 (1-2): 3–24  (French).
  8. ^ Pomerance, Carl (2008), "Computational Number Theory", The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362 
  9. ^ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf
  10. ^ Andrzej Dąbrowski (1996), "On the diophantine equation x! + A = y2", Nieuw Archief voor Wiskunde, IV. 14: 321–324 
  11. ^ Stewart, C. L.; Tijdeman, R. (1986), "On the Oesterlé-Masser conjecture", Monatshefte für Mathematik 102 (3): 251–257, doi:10.1007/BF01294603. 
  12. ^ Stewart, C. L.; Kunrui Yu (1991), "On the abc conjecture", Mathematische Annalen 291 (1): 225–230, doi:10.1007/BF01445201. 
  13. ^ Stewart, C. L.; Kunrui Yu (2001), "On the abc conjecture, II", Duke Mathematical Journal 108 (1): 169–181, doi:10.1215/S0012-7094-01-10815-6. 
  14. ^ "Synthese resultaten", RekenMeeMetABC.nl, http://www.rekenmeemetabc.nl/?item=h_stats, retrieved January 1, 2011  (Dutch).
  15. ^ "Data collected sofar", ABC@Home, http://abcathome.com/data/, retrieved March 20, 2011 
  16. ^ "100 unbeaten triples". Reken mee met ABC. 2010-11-07. http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=1. 
  17. ^ J. Browkin, J. Brzeziński (1994), "Some remarks on the abc-conjecture", Math. Comp. (American Mathematical Society) 62 (206): 931–939, doi:10.2307/2153551, JSTOR 2153551 

Further reading

  • Browkin, Jerzy (2000), "The abc-conjecture", in Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J., Number Theory, Trends in Mathematics, Basel: Birkhäuser, pp. 75–106, ISBN 3764362596 
  • Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008), The Princeton Companion to Mathematics, Princeton: Princeton University Press, pp. 361–362, 681, ISBN 978-0691118802 
  • Guy, Richard K. (2004), Unsolved Problems in Number Theory, Berlin: Springer-Verlag, ISBN 0387208607 
  • Lando, Sergei K.; Zvonkin, Alexander K. (2004), "Graphs on Surfaces and Their Applications", Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II (Springer-Verlag) 141, ISBN 3540002030 

External links


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