## Functiones et Approximatio Commentarii Mathematici

### $abc$ triples

#### Abstract

The $abc$ conjecture, one of the most famous open problems in number theory, claims that three relatively prime positive integers $a,b,c$ satisfying $a+b=c$ cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than $c$. Triples of relatively prime numbers satisfying $a+b=c$ are called {\em $abc$ triples} if the product of their distinct prime divisors is strictly less than $c$. We catalog what is known about $abc$ triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we collect motivations and heuristics supporting the $abc$ conjecture, as well as some of its refinements and generalizations, and we describe the state-of-the-art progress towards establishing the conjecture.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 55, Number 2 (2016), 145-176.

Dates
First available in Project Euclid: 17 December 2016

https://projecteuclid.org/euclid.facm/1481943880

Digital Object Identifier
doi:10.7169/facm/2016.55.2.2

Mathematical Reviews number (MathSciNet)
MR3584566

Zentralblatt MATH identifier
06862559

#### Citation

Martin, Greg; Miao, Winnie. $abc$ triples. Funct. Approx. Comment. Math. 55 (2016), no. 2, 145--176. doi:10.7169/facm/2016.55.2.2. https://projecteuclid.org/euclid.facm/1481943880

#### References

• ABC@Home, The algorithm, Mathematisch Instituut Universiteit Leiden. http://abcathome.com/algorithm.php (accessed August 3, 2014)
• D. P. Anderson, R. Walton, C. Fenton, et al., Berkeley Open Infrastructure for Network Computing (BOINC), April 10, 2002. http://boinc.berkeley.edu (accessed August 3, 2014)
• T.M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.
• A. Baker, Logarithmic forms and the $abc$-conjecture, in Number Theory: Diophantine, Computational and Algebraic Aspects, Eger–1996, K. Györy, A. Pethö, and V.T. Sós, Proc. Intl. Conf., de Grutyer, Berlin, 1998, 37–44.
• E. Bombieri, Roth's theorem and the abc conjecture, preprint (1994).
• D. Boyd, G. Martin, and M. Thom, Squarefree values of trinomial discriminants, LMS J. Comput. Math. 18 (2015), no. 1, 148–169.
• J. Browkin and J. Brzezinski, Some remarks on the $abc$-conjecture, Math. Comp. 62 (1994), no. 206, 931–939.
• N.A. Carella, Probable counterexamples of the ABC conjecture, 2003. http://arxiv.org/pdf/math/0503401v1.pdf
• J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, 1997.
• B. de Smit, Update on $abc$ triples, March 2007. http://pub.math.leidenuniv.nl/~smitbde/abc (accessed August 3, 2014)
• T. Dokchitser, LLL and ABC, J. Number Theory 107 1 (2004), 161–167.
• N. Elkies, $ABC$ implies Mordell, Int. Math. Res. Notices (1991), no. 7, 99–109.
• J.S. Ellenberg, Congruence $ABC$ implies $ABC$, Indag. Math. N.S. 11 (2), 197–200 (2000).
• A. Granville and H.M. Stark, ABC implies no “Siegel zeros” for $L$-functions of characters with negative discriminant, Invent. Math. 139, no.3, 509–523.
• A. Granville and T.J. Tucker, It's as easy as $abc$, Notices of the A.M.S. 49 (2002), 1224–1231.
• B. Gross and D. Zagier, On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220.
• G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1979.
• C. Hurlburt, Good $abc$ triples. http://www.phfactor.net/abc (accessed August 3, 2014)
• joro, Argument againts the $abcd$ conjecture with extra gcd conditions, MathOverflow. https://mathoverflow.net/questions/185857/argument-againts-the-abcd-conjecture-with-extra-gcd-conditions (accessed January 10, 2015)
• A. Klenke, Probability Theory, Springer, 2014.
• S.K. Lando and A.K. Zvonkin, Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II 141 (Springer-Verlag), 2004.
• S. Lang, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 37–75.
• S. Lang, Undergraduate Algebra, Springer, New York, 2005.
• K. Lauter and B. Viray, On singular moduli for arbitrary discriminants, Int. Math. Res. Not. IMRN, to appear.
• A.K. Lenstra, H.W. Lenstra, Jr., and L. Lovász, Factoring polynomials with rational coefficients, Mathematische Annalen 261 (1982), no. 4, 515–534.
• Mabcp, MarioKappa, and T. Tao, ABC conjecture, Polymath, August 8, 2009. http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture (accessed August 11, 2014)
• R.C. Mason, Diophantine equations over function fields, London Mathematical Society Lecture Note Series, 96, Cambridge University Press, Cambridge, 1984.
• D.W. Masser, Open problems, Proc. Symp. Analytic Number Theory, (W.W.L. Chen, ed.), Imperial Coll. London, 1985.
• M.R. Murty and H. Pasten, Modular forms and effective Diophantine approximation, Journal of Number Theory 133 (2013), no. 11, 3739–3754.
• J. Neukirch, Algebraic Number Theory (translated by N. Schappacher), Springer–Verlag, Berlin, 1999.
• A. Nitaj, http://www.math.unicaen.fr/~nitaj/abc.html#Consequences, 2013.
• I. Niven, H.S. Zuckerman, and H.L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, John Wiley & Sons, Inc., New York (1991).
• J. Oesterlé, Nouvelles approches du théorème de Fermat, Sém. Bourbaki 1987-1988, Vol.694, Astérisque 161-162 (1988), 165–186.
• C. Pomerance, Computational number theory, in Princeton Companion to Mathematics (ed. W.T. Gowers), Princeton U. Press, Princeton, New Jersey, 2008, 348–362.
• Reken mee met ABC, Mathematisch Instituut Universiteit Leiden. http://www.rekenmeemetabc.nl (accessed August 3, 2014)
• O. Robert, C.L. Stewart, and G. Tenenbaum, A refinement of the abc conjecture, Bull. London Math. Soc. 46 (2014), no. 6, 1156–1166.
• J.H. Silverman, Arithmetic of Elliptic Curves, Springer, New York, 2009.
• J.H. Silverman, Wieferich's criterion and the $abc$-conjecture, J. Number Theory 30 (1988), 226–237.
• K. Soundararajan, The distribution of prime numbers, in Equidistribution in number theory, an introduction, NATO Sci. Ser. II Math. Phys. Chem. 237, Springer (Dordrecht), 2007, 59–83.
• C.L. Stewart, A note on the product of consecutive integers, in Topics in classical number theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. János Bolyai 34, North-Holland, Amsterdam, 1984, 1523–1537.
• C.L. Stewart and R. Tijdeman, On the Oesterlé-Masser conjecture, Monatsh. Math. 102 (1986), no.3, 251–257.
• C.L. Stewart and K. Yu, On the $abc$ conjecture, Math. Ann. 291 (1991), 225–230.
• C.L. Stewart and K. Yu, On the $abc$ conjecture, II, Duke Math. Journal 108 (2001), 169–181.
• W.W. Stothers, Polynomial identities and Hauptmoduln, Q.J. Math. Oxford 32 (1981), 349–370.
• T. Tao, The probabilistic heuristic justification of the ABC conjecture, What's new, September 18, 2012. http://terrytao.wordpress.com/2012/09/18/ the-probabilistic-heuristic-justification-of-the-abc-conjecture/ (accessed August 11, 2014)
• W.P. Thurston, On proof and progress in mathematics, Bulletin of the AMS 30 (1994), no. 2, 161–177.
• J. van der Horst, Finding ABC-triples using elliptic curves , Masters thesis, Universiteit Leiden, 2010.
• M. van Frankenhuysen, A lower bound in the $abc$ conjecture, J. Number Theory 82 (2000), 91–95.
• P. Vojta, Diophantine Approximations and Value Distribution Theory, Springer, New York, 1987.
• P. Vojta, A more general ABC conjecture, Internat. Math. Res. Notices 21 (1998), 1103–1116.
• L.C. Washington, Elliptic Curves: Number theory and cryptography, 2nd ed., Chapman & Hall/CRC, Boca Raton, 2008.