The abc conjecture is a mathematical conjecture made by Joseph Oesterlé and David Masser in 1985 . It is about the shared content on prime factors of triples each other relatively prime natural numbers where the third is the sum of the two others. It describes in precise form the phenomenon that the product of all the different prime factors occurring in such a triple is generally not or only slightly smaller than the largest number of the triple. The additive relationship of a triple forces a strong restriction on the multiplicative structure of the triple numbers.
Heuristically , the abc assumption is based on the fact that natural numbers with a large number of multiple prime factors - so-called highly potent or "rich" numbers - occur comparatively rarely. Based on a definition by Barry Mazur , a natural number can be described as multiplicative highly potent if its binary representation is significantly longer than the binary representation of its greatest square-free divisor , i.e. the product of all the various prime factors it contains. Then the abc conjecture for two coprime high-power numbers and says that neither their sum nor their difference can be high-potent, possibly with exceptions if is small.
The conjecture has not yet been proven or refuted, but because of its difficulty, and even more because of its importance, it is a prominent successor to the solved Fermat's conjecture (new "Holy Grail"). Dorian Goldfeld even described it as the most important unsolved problem of Diophantine analysis . A large number of far-reaching number theoretic statements are already known which would follow from the validity of the abc conjecture.
A triple is called abc -triple if and are relatively prime positive integers and their sum is. Due to elementary properties of the divisibility relationship , both to and to are coprime.
The radical of a positive integer is the product of the different prime factors of . Prime factors that appear multiple times in the prime factorization of are only taken into account once when calculating . For example is
The inequality holds for an abc triple
so it is called an abc hit . Examples are (1, 8, 9), (5, 27, 32), (32, 49, 81) and the triple found by Éric Reyssat with , for which the quotient log ( c ) / log (rad ( abc )) = 1.62991 ... is particularly large. abc hits are rare. Among the 15.2 million abc triples with there are only 120 abc hits and among the 380 million abc triples with there are 276. In 2006, Sander Dahmen proved a lower estimate for the number of abc hits up to a given limit and confirmed that there are infinitely many, but his formula only predicts about a million abc hits below .
The worldwide project ABC @ Home has so far been able to find over 33.18 million abc hits through distributed computing and has set itself the goal of creating a complete list of all abc hits for . The list for was finalized in November 2011. The project was made possible by the programming of an algorithm that reduced the effort required to determine all abc hits from the obvious proportional to to almost proportional to calculation steps.
Masser proved that the ratio can be arbitrarily small, although it is usually greater than 1. For exponents (even if they come arbitrarily close to 1), however, he formulated with Oesterlé the abc conjecture that it has a positive lower bound.
More precisely, the abc presumption reads :
- For every real one there is a constant , so that the following inequality holds for all triples of relatively prime positive integers with :
The presumption is formulated for because it is demonstrably wrong for as mentioned.
You can also formulate the conjecture for any positive or negative whole numbers and then only have to replace the inequality with on the left-hand side .
Another, equivalent formulation of the conjecture is given below.
Any number of abc hits with the same b
For and a number of the form there are numbers that form an abc hit with. With that, one chooses with , so that applies.
The property of the triples to be abc hits can be shown as follows. First is
- , so .
If the congruences are calculated modulo 49, one obtains
So is and .
Consequences and variants of the abc presumption
Conclusions from the abc assumption
The presumption has not yet been proven, but it has a number of interesting consequences. Many solved and unsolved Diophantine problems can be deduced from this conjecture. In particular, the very complex proof of Fermat's Great Theorem would be reduced to one page. The sentences or assumptions that would result from a proof of the abc presumption include:
- Thue-Siegel-Roth theorem , as Machiel van Frankenhuysen showed in 1999.
- Great Fermatsch sentence
- Mordell's conjecture ( proven by Gerd Faltings ), as Noam Elkies showed in 1991. The conjecture asserts the finiteness of the number of points on an algebraic curve of gender greater than 1 over a number field K. The abc conjecture even implies a limit for the size (more precisely the so-called height) of the points on the curves over K (depending on the constant occurring in the abc assumption). The abc presumption thus provides an effective version of the murder presumption, in contrast to the evidence known to date.
- Erdős-Woods conjecture (M. Langevin 1993)
- Catalan conjecture
- Fermat-Catalan conjecture
- the existence of an infinite number of non- Wieferich prime numbers . General showed Joseph Silverman in 1988 that from the abc follows presumption that it , infinitely many prime numbers are, for not divisible.
- the weak form of the Hall conjecture, which provides an asymptotic lower bound for the amount of the difference between cubic numbers and square numbers.
- the conjecture of Lucien Szpiro (an inequality between leader and discriminant of elliptical curves over the rational numbers). This assumption is even equivalent to the abc assumption . More precisely, it is the generalized Szpiro conjecture (see below).
- the Pillai conjecture from SS Pillai .
- an effective form of Siegel's theorem about integer points on algebraic curves.
Szpiro's conjecture in the theory of elliptic curves follows from the abc conjecture, as Oesterlé and Nitaj showed. The conjecture is as follows: For each there is a constant so that for each elliptic curve with a minimal discriminant and leader it applies:
The generalized Szpiro conjecture, which is equivalent to the abc conjecture, reads: For every and there is a constant such that for all integers for which and the largest prime factor of , is less than or equal :
As an example, the abc conjecture is applied to Fermat's great theorem that
has no solution in positive integers (which are assumed to be relatively prime) for
If you insert and use in the inequality of the abc conjecture
the inequality then reads:
Substituting in this inequality by , then one has for an upper bound for z:
This means that the Fermat equation can only have a finite number of solutions and from a certain value of the exponent , which only depends on, that would be given by the abc conjecture, no solution at all, there . One need only check all cases up to this limit with other methods in order to prove the Fermat conjecture (for a large number of exponents the correctness of the conjecture was known before Andrew Wiles' proof).
Special forms of the abc presumption and weak abc presumption
In 1996 Alan Baker proposed a tightening of the conjecture and made it more precise in 2004. While the total size of the multiplicative building blocks characterizes the numbers involved in the triplet, the number of their different prime factors is a measure of their level of detail. Baker combined both measures and arrived at an abc conjecture with an absolute, independent, constant
If you take into account that the right-hand side has a minimum around at , and after replacing it down with in the denominator , you get a free version
- , an absolute constant.
Andrew Granville noted that the last factor is almost equivalent to Θ (r), the number of natural numbers up to r that are only divisible by prime factors of r . This gives his conjecture to
- , an absolute constant.
A study of the 196 extremal abc triples known at the time showed that presumably and can be chosen. The second value may have to be modified slightly based on more recent numerical results.
There are also weaker forms of the abc conjecture that one tries to prove. If in the original formulation of the abc conjecture and equals 1, one has a variant of the weak abc conjecture (with the same requirements for the abc triples as above):
From this variant it follows immediately (through a similar argumentation as above) the validity of the Fermat conjecture for powers greater than five. More generally, the weak abc presumption is often introduced via a slightly different formulation of the abc presumption.
Let the quality (also power , abc-ratio ) of an ( a , b , c ) -triple, thus the solution of with and thus a measure of the excess of c over the common “prime number content” r of the triple. Extensive numerical search, for example in the ABC @ Home project, has so far yielded a maximum value of around 1.63 for q (found by Eric Reyssat, see above). In total, only 241 abc triples with a quality> 1.4 could be detected in 34 years . The actual abc conjecture, also known as the strong abc conjecture , then states that
has only a finite number of solutions for an arbitrary one .
The value 1 is the best possible lower limit for d . If one sets d = 1, there are infinitely many solutions. But even an arbitrarily small value above 1, according to the strong abc conjecture, means that the number of solutions is finite.
The weak abc conjecture says that q has an upper bound. In the special case given above, the upper bound 2 was assumed. The strong abc conjecture follows the validity of the weak abc conjecture, but not vice versa.
In a symmetrical form, the conjecture can also be expressed as a statement of the ratio of the height H ( a , b , c ) = max (| a |, | b |, | c |), which measures the size of the numbers involved, to the radical R ( a , b , c ), which measures the prime content. Then the strong abc conjecture says that for each there are only finitely many coprime solutions a , b , c with:
Jeffrey Lagarias and Kannan Soundararajan set of abc presumption an " xyz presumption" to the side in the event that all prime factors of the radical of a triplet by a small constant S ( smoothness , smoothness ) are restricted, that is . It says that there are only finitely many abc triples with .
B. de Weger determined in the results of the ABC @ Home project that triple with S = 43 and (presumably) the largest z as
Conrey, Holmstrom and McLaughlin found it to be a triplet with the maximum smoothness index
More reviews of an abc hit
As early as 1986, Cameron L. Stewart and Robert Tijdeman showed that the "quality" assessment of the abc hits (with the designations and , )
cannot converge too quickly to 1 for growing and thus again that there is no for . They proved the existence of an infinite number of abc triples with
- for each .
In 2000 Machiel van Frankenhuysen tightened this statement with that suggests to investigate whether a given triple with the rating
the barrier exceeds or not, and to analyze the distribution of detected extremal examples. The following theoretical (heuristic) considerations suggest that this evaluation can be infinitely large based on the set of abc hits.
Of proven performance on the distribution of natural numbers with (justified and often confirmed, but unproven) below a given barrier and made assumptions about the randomness of the prime factorization of natural numbers in unstructured amounts could van Frankenhuysen the stricter lower estimate with a smaller denominator
- applies infinitely often
derive. Depending on the approach you can choose one or one , that could not be clarified. The second variant was also found by CL Stewart and G. Tenenbaum (2007, cf.) and tightened with Olivier Robert in 2014. A simple transformation results in the elegant evaluation "merit"
as a squared analogue of the target test variable .
The current world record triple with regard to both ratings with and was discovered by Ralf Bonse on October 28, 2011 and is
- , ( is obviously not highly potent as a multiplicative)
Of particular interest are those abc triples that limit the decrease in quality with increasing amount from below. An abc triple is called "unbeaten" (meaning "unsurpassed") if every known abc triple with a larger one has a lower quality.
abc guess for polynomials
In 1983 Wilson Stothers and Richard Mason independently proved the following, hitherto unknown theorem for polynomials:
Let coprime, non-constant polynomials with . Then
where the number of different zeros is from . In a sense, this is the “function body” analog of the abc conjecture. Its proof is relatively simple and, as in the case of the abc conjecture, it follows e.g. B. Fermat's theorem for polynomials. The translation from the polynomial case into the abc conjecture for integers is done by setting, where the product of the "prime factors" of is, extends over all roots of , and the degree is replaced by its analogue, the logarithm (da ) .
However, this “model” version of the abc conjecture was not the direct motivation for the conjecture by Oesterlé and Masser. The motive for the conjecture did not result from numerical calculations, but rather from deep-seated studies of elliptic curves in number theory, which are partly reflected in the related conjecture by Lucien Szpiro (see above).
So far the following inequalities have been proven for c and rad ( abc ):
1986, CL Stewart and R. Tijdeman:
1991, CL Stewart and Kunrui Yu:
1996, CL Stewart and Kunrui Yu:
where C 1 is a fixed constant and C 2 and C 3 are positive, easily calculated constants depending on ε.
Attempts to prove
In August 2012, Shin'ichi Mochizuki released possible evidence that is currently being investigated. Mochizuki started from Szpiro's conjecture about elliptic curves, which is equivalent to the abc conjecture, and applied extensive concepts and methods that were only recently developed by him and so far only a few known. In March 2015, a twelve-day workshop on the Inter-Universale Teichmüller Theory was held at its institute in Kyoto , and the Clay Mathematics Institute held another five-day workshop in December 2015. However, even six years after its publication, the proof did not convince most specialists, and its correctness has been doubted by prominent mathematicians. Jakob Stix and Peter Scholze announced in 2018 that they had identified a fundamental gap in the evidence of Mochizuki. Mochizuki continues to hold on to his evidence. On April 3, 2020, Nature reported that its 600-page evidence had been accepted for publication by the journal Publications of the RIMS . Scholze informed Nature in an email that his criticism of the evidence had not changed. The publication is essentially unchanged compared to the preprints and only takes into account the criticism of Scholze and Stix in a few comments. The FAZ commented that the “official publication of the evidence” was “an unheard of process ” despite the technical criticism that had not been cleared up , and that “the validity of a piece of important mathematics was now a question of taking it for granted” .
- ABC @ Home (project for distributed computing)
- Enrico Bombieri , Walter Gubler: Heights in Diophantine Geometry. Cambridge University Press, 2006, chapter 12
- DW Masser: Open problems. In: WWL Chen (Ed.): Proc. Symp. Analytic Number Theory. Imperial College, London 1985. The conjecture is formulated there for the first time.
- CL Stewart, R. Tijdeman: On the Oesterlé-Masser Conjecture. Monthly books for mathematics 102 (1986), pp. 251-257
- Joseph Oesterlé: Nouvelles approches du “théorème” de Fermat. Séminaire Bourbaki No. 694, 1987/8
- DW Masser: Note on a conjecture of Szpiro. In: Astérisque. Volume 184, 1990. p. 19.
- Richard Kenneth Guy : Unsolved Problems in Number Theory. Springer-Verlag, Berlin 2004, ISBN 0-387-20860-7 .
- Gerhard Frey : The ABC conjecture. In: Spectrum of Science Dossier: "The greatest riddles of mathematics". Issue 6/2009, ISBN 978-3-941205-34-5 , pages 48-55.
- Matthias Mahl: Construction of good ABC triples with the LLL algorithm . Grin-Verlag, Munich 2010, ISBN 3-640-68185-1 .
- Rob Tijdeman: Het abc vermoeden , Nieuw Archief voor Wiskunde, December 2015, pdf (PDF)
- Michel Waldschmidt : Lecture on the abc conjecture and some of its consequences , in: Pierre Cartier, ADR Choudhary, Michel Waldschmidt (ed.), Mathematics in the 21st century, Springer 2015, pp. 211-230
- Abderrahmane Nitaj's website about the abc presumption
- Ivars Peterson: The Amazing ABC Conjecture ( Memento from June 28, 2013 in the Internet Archive ) December 8, 1997 (English).
- Eric W. Weisstein : abc Conjecture . In: MathWorld (English).
- Andrew Granville, Thomas J. Tucker: It's As Easy As abc. (PDF) Notices AMS, 2002.
- ABC at Home project that uses distributed computing to collect data to better understand the problem
- Serge Lang : The abc assumption . In: Elements of Mathematics , Volume 48, 1993.
- Frits Beukers: presentation slides (PDF, 527 kB; English)
- Masser and Oesterlé on the origin of the abc assumption. (Dutch) with links to Masser, Oesterlé
- Michel Waldschmidt : On the abc conjecture and some of its consequences. (PDF) presentation
- Marlene Weiß: Mathematicians have been trying unsuccessfully to understand a proof for four years . In: Süddeutsche Zeitung
- ↑ Noam Elkies: The ABC's of Number Theory (PDF; 417 kB)
- ^ The Amazing ABC Conjecture
- ↑ Gerhard Frey: The ABC conjecture . Spectrum d. Knowledge February 2009, pp. 70-77
- ^ Sander Roland Dahmen: Lower bounds for numbers of ABC hits . (PDF; 113 kB) In: Journal Number Theory , 128, 2008, No. 6, pp. 1864–1873
- ↑ Data collected sofar ( Memento from April 24, 2010 in the Internet Archive )
- ↑ Synthesis results ( Memento of March 3, 2012 in the Internet Archive )
- ↑ Willem Jan Palenstijn: Enumerating ABC triples . ( Memento of February 3, 2014 in the Internet Archive ; PDF; 816 kB)
- ↑ A simple proof by Wojtek Jastrzebowski and Dan Spielman elements found in Lang, mathematics, Vol. 48 1993, p 94. Your counterexample to the abc conjecture with is . One proves by induction that b is divisible by . This results in an inequality that cannot be satisfied for all k.
- ↑ Robin Weezepoel ( Memento from July 27, 2014 in the Internet Archive )
- ↑ Machiel van Frankenhuysen: The ABC conjecture implies Roth's theorem and Mordell's conjecture , Matemática Contemporânea, Volume 16, 1999, pp. 45-72
- ↑ William Stein: Szpiro and ABC ( Memento from February 17, 2009 in the Internet Archive ) (English)
- ↑ arxiv : math / 0408168 Andrea Surroca, Siegel's theorem and the abc conjecture, Riv. Mat. Univ. Parma (7) 3, 2004, pp. 323-332
- ↑ Waldschmidt, Lecture on the abc conjecture and some of its consequences, in: Cartier u. a., Mathematics in the 21st century, Springer 2015, p. 214
- ^ Alan Baker: Logarithmic forms and the abc-conjecture. In: Györy, Pethö, T. Sos (ed.) Number Theory, Eger 1996. , de Gruyter 1998, pp. 37-44. Experiments on the abc-conjecture. Publ. Math. Debrecen 65 (2004), pp. 253-260
- ↑ for example Lukas Pottmeyer: The density of square-free values of integer polynomials , diploma thesis, University of Dortmund, 2009, page III, PDF file (PDF)
- ↑ Bart de Smit: Update on ABC triples (also further numerical results)
- ↑ ABC at Home website ( Memento from November 18, 2009 in the Internet Archive )
- ↑ Lagarias, Soundararajan: Smooth solutions of the abc conjecture . In: J. Theory Nombres Bordeaux , Volume 23, 2011, p. 209, arxiv : 0911.4147 Preprint
- ↑ Smooth Solutions to the Equation A + B = C ( Memento from December 26, 2015 in the Internet Archive ) (PDF; 237 kB) Preprint 2010.
- ↑ Benne de Weger: Numerical data related to the Lagarias-Soundararajan xyz-conjecture. (PDF; 381 kB) revised preprint 2012.
- ↑ JB Conrey, MA Holmstrom, TL McLaughlin: Smooth Neighbors , Experimental Mathematics 22 (2013), pp. 195-202
- ^ Machiel van Frankenhuysen: A lower bound in the abc conjecture. J. Number Theory, 82 (2000), pp. 91-95
- ↑ Machiel van Frankenhuysen: Hyperbolic spaces and the abc conjecture . Dissertation Nijmegen 1995
- ↑ Carl Pomerance: Computational Number Theory (review article, pdf; 249 kB)
- ^ O. Robert, CL Stewart, G. Tenenbaum: A refinement of the abc conjecture . (Preprint, pdf; 322 kB)
- ↑ Bart de Smit / ABC triples / by merit
- ↑ Bart de Smit / ABC triples / unbeaten
- ^ R. Mason: Diophantine equations over function fields. Cambridge University Press 1984
- ^ WW Stothers: Polynomial identities and main modules. Quarterly Journal Mathematics, Oxford, II. Ser., Vol. 32, 1981, pp. 349-370. Joseph Silverman also independently proved the theorem, which is also called the PQR theorem or Stothers-Mason (Silverman) theorem.
- ↑ see e.g. B. Serge Lang: Elements of Mathematics. Volume 48 (1993), pp. 91f
- ↑ Oesterlé on the motivation behind her postulation of the abc presumption ( memento from March 4, 2016 in the Internet Archive )
- ^ Mochizuki: Inter-Universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations , Preprint August 2012, online on his homepage
↑ Holger Dambeck: Japanese presents solution for prime number riddles , Spiegel-Online, September 26, 2012
Philip Ball : Proof claimed for deep connection between prime numbers , Nature News, September 10, 2012
Caroline Chen: The paradox of the proof. (As of May 9, 2013)
Peter Woit: Latest on abc . (As of December 19, 2013)
- ^ IUT Theory of Shinichi Mochizuki at the CMI
- ^ Biggest mystery in mathematics in limbo after cryptic meeting . Nature
- ^ Frank Calegari: The ABC conjecture has (still) not been proved . 17th December 2017
- ↑ Erica Klarreich: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture . Quanta Magazine, September 20, 2018
- ↑ Mochizuki website with the report by Scholze and Stix and answers from Mochizuki
- ↑ a b Ulf von Rauchhaupt : ABC conjecture: number theory at the limit . In: faz.net . April 9, 2020, ISSN 0174-4909 ( faz.net [accessed April 10, 2020]).
- ↑ Shinichi Mochizuki: March 2018 Discussions on IUTeich. Retrieved April 10, 2020 .
- ↑ Davide Castelvecchi: Mathematical proof that rocked number theory will be published . In: Nature . tape 580 , April 3, 2020, p. 177–177 , doi : 10.1038 / d41586-020-00998-2 ( nature.com [accessed April 10, 2020]).