abc conjecture
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 August 2014.
Formulations
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
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
holds.
A third formulation of the conjecture involves the quality, q(a, b, c), of the triple (a, b, c), defined by:
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
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). Since b=32r for some r, 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
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.
- Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers
- Fermat's Last Theorem for all sufficiently large exponents (already proven in general by Andrew Wiles) (Granville 2002)
- The Mordell conjecture (already proven in general by Gerd Faltings) (Elkies 1991)
- The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)
- The existence of infinitely many non-Wieferich primes (Silverman 1988)
- The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers (Nitaj 1996)
- The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers (Pomerance 2008)
- 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) (Granville 2000)
- P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.[2]
- A generalization of Tijdeman's theorem 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.
- It is equivalent to 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 for all coprime integer x,y, the radical of f(x,y) exceeds C.max{|x|,|y|}n-β.[3][4]
- It is equivalent to the modified Szpiro conjecture, which would yield a bound of rad(abc)1.2+ε (Oesterlé 1988).
- Dąbrowski (1996) has shown that the abc conjecture implies that the Diophantine equation n! + A= k2 has only finitely many solutions for any given integer A.
- There are ~cfN positive integers n ≤ N for which f(n)/B' is squarefree, with cf > 0 a positive constant defined as . (Granville 1998)
Fermat's Last theorem
Fermat's last theorem was proven by Andrew Wiles and the proof is famous for its difficulty. But if 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
( ).
Now we get
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]
Theoretical results
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
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.
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[update], 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]
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
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 over occurs when .
This incited Baker (2004) to propose a sharper form of the abc conjecture, namely:
with an absolute constant. After some computational experiments in order to find a value for , he found that a value of 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
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.
Attempts at solution
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
Notes
- ^ Note that when it is given that 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.
- ^ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf
- ^ Mollin (2009)
- ^ Mollin (2010) p.297
- ^ Granville, Andrew; Tucker, Thomas (2002). “It’s As Easy As abc”. Notices of the AMS 49 (10): 1224–1231.
- ^ Yamazaki, Takao"Fermat's Last Theorem and ABC conjecture"(Japanese), Sugaku seminar Oct 2010.
- ^ "Synthese resultaten", RekenMeeMetABC.nl, retrieved October 3, 2012 Template:Nl icon.
- ^ "Data collected sofar", ABC@Home, retrieved April 30, 2014
- ^ "100 unbeaten triples". Reken mee met ABC. 2010-11-07.
- ^ "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.
- ^ 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
- ^ Ball, Phillip (10 September 2012), "Proof claimed for deep connection between primes", Nature.
- ^ Cipra, Barry (September 12, 2012), "ABC Proof Could Be Mathematical Jackpot", Science.
- ^ The Paradox of the Proof
- ^ Kevin Hartnett (3 November 2012). "An ABC proof too tough even for mathematicians". Boston Globe.
- ^ [1]
References
- Baker, Alan (1998). "Logarithmic forms and the abc-conjecture". In Győry, Kálmán (ed.). Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. pp. 37–44. ISBN 3-11-015364-5. Zbl 0973.11047.
{{cite book}}
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(help) - Baker, Alan (2004). "Experiments on the abc-conjecture". Publ. Math. Debrecen. 65: 253–260.
{{cite journal}}
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(help) - Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
- Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. doi:10.2307/2153551. JSTOR 2153551.
{{cite journal}}
: Invalid|ref=harv
(help) - Browkin, Jerzy (2000). "The abc-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 75–106. ISBN 3-7643-6259-6.
{{cite book}}
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(help) - Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y2". Nieuw Archief voor Wiskunde, IV. 14: 321–324. Zbl 0876.11015.
{{cite journal}}
: Invalid|ref=harv
(help) - Elkies, N. D. (1991). "ABC implies Mordell". Intern. Math. Research Notices. 7 (7): 99–109. doi:10.1155/S1073792891000144.
{{cite journal}}
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(help)CS1 maint: unflagged free DOI (link) - Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons (September): 26–34. JSTOR 25678079.
{{cite journal}}
: Invalid|ref=harv
(help) - Goldfeld, Dorian (2002). "Modular forms, elliptic curves and the abc-conjecture". In Wüstholz, Gisbert (ed.). A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press. pp. 128–147. ISBN 0-521-80799-9. Zbl 1046.11035.
{{cite book}}
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(help) - Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 361–362, 681. ISBN 978-0-691-11880-2.
{{cite book}}
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(help) - Granville, A. (1998). "ABC Allows Us to Count Squarefrees" (PDF). International Mathematics Research Notices. 1998: 991–1009. doi:10.1155/S1073792898000592.
{{cite journal}}
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(help)CS1 maint: unflagged free DOI (link) - Granville, Andrew; Stark, H. (2000). "ABC implies no "Siegel zeros" for L-functions of characters with negative exponent" (PDF). Inventiones Mathematicae. 139: 509–523. doi:10.1007/s002229900036.
{{cite journal}}
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(help) - Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231.
{{cite journal}}
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(help) - Guy, Richard K. (2004). Unsolved Problems in Number Theory. Berlin: Springer-Verlag. ISBN 0-387-20860-7.
{{cite book}}
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(help) - Lando, Sergei K.; Zvonkin, Alexander K. (2004). Graphs on Surfaces and Their Applications. Vol. 141. Springer-Verlag. ISBN 3-540-00203-0.
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(help) - Langevin, M. (1993). "[Cas d'égalité pour le théorème de Mason et applications de la conjecture abc] Error: {{Lang}}: text has italic markup (help)". Comptes rendus de l'Académie des sciences. 317 (5): 441–444.
{{cite journal}}
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(help) Template:Fr icon - Masser, D. W. (1985). "Open problems". In Chen, W. W. L. (ed.). Proceedings of the Symposium on Analytic Number Theory. London: Imperial College.
- Mollin, R.A. (2009). "A note on the ABC-conjecture" (PDF). Far East J. Math. Sci. 33 (3): 267–275. ISSN 0972-0871. Zbl 1241.11034.
- Mollin, Richard A. (2010). Advanced number theory with applications. Boca Raton, FL: CRC Press. ISBN 978-1-4200-8328-6. Zbl 1200.11002.
- Nitaj, Abderrahmane (1996). "[La conjecture abc] Error: {{Lang}}: text has italic markup (help)". Enseign. Math. 42 (1–2): 3–24.
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(help) Template:Fr icon - Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR992208
- Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.
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(help) - 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. Zbl 0654.10019.
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(help) - Robert; Stewart; Tenenbaum (2014). " "A refinement of the abc conjecture". preprint.
- Robert; Tenenbaum (2013). "Sur la répartition du noyau d'un entier". Indag. Math. 24: 802–914.
- Stewart, C. L.; Tijdeman, R. (1986). "On the Oesterlé-Masser conjecture". Monatshefte für Mathematik. 102 (3): 251–257. doi:10.1007/BF01294603.
{{cite journal}}
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(help) - Stewart, C. L.; Yu, Kunrui (1991). "On the abc conjecture". Mathematische Annalen. 291 (1): 225–230. doi:10.1007/BF01445201.
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(help) - Stewart, C. L.; Yu, Kunrui (2001). "On the abc conjecture, II". Duke Mathematical Journal. 108 (1): 169–181. doi:10.1215/S0012-7094-01-10815-6.
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(help)
External links
- ABC@home Distributed Computing project called ABC@Home.
- Easy as ABC: Easy to follow, detailed explanation by Brian Hayes.
- Weisstein, Eric W. "abc Conjecture". MathWorld.
- Abderrahmane Nitaj's ABC conjecture home page
- Bart de Smit's ABC Triples webpage
- http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
- The amazing ABC conjecture
- The ABC's of Number Theory by Noam D. Elkies
- Questions about Number by Barry Mazur
- Philosophy behind Mochizuki’s work on the ABC conjecture on MathOverflow
- ABC Conjecture Polymath project wiki page linking to various sources of commentary on Mochizuki's papers.
- abc Conjecture Numberphile video