Editing Abc conjecture (section)

==Some consequences==

The ''abc'' conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a [[conditional proof]]. The consequences include:

* [[Roth's theorem]] on [[Diophantine approximation]] of [[algebraic number]]s.{{sfnp|Bombieri|1994|p={{page needed|date=January 2022}}}}{{sfn|Waldschmidt|2015}}
* The [[Faltings's theorem|Mordell conjecture]] (already proven in general by [[Gerd Faltings]]).{{sfnp|Elkies|1991}}
* As equivalent, [[Vojta's conjecture]] in dimension 1.{{sfnp|Van Frankenhuijsen|2002}}
* The [[Erdős–Woods number|Erdős–Woods conjecture]] allowing for a finite number of counterexamples.{{sfnp|Langevin|1993}}
* The existence of infinitely many non-[[Wieferich prime]]s in every base ''b'' > 1.{{sfnp|Silverman|1988}}
* The weak form of [[Marshall Hall's conjecture]] on the separation between squares and cubes of integers.{{sfnp|Nitaj|1996}}
* [[Fermat's Last Theorem]] has [[Wiles's proof of Fermat's Last Theorem|a famously difficult proof by Andrew Wiles]]. However it follows easily, at least for <math>n \ge 6</math>, from an effective form of a weak version of the ''abc'' conjecture. The ''abc'' conjecture says the [[Limit superior and limit inferior|lim sup]] of the set of all qualities (defined above) is 1, which implies the much weaker assertion that there is a finite upper bound for qualities. The conjecture that 2 is such an upper bound suffices for a very short proof of Fermat's Last Theorem for <math>n \ge 6</math>.<ref>{{cite journal |last1=Granville |first1=Andrew |last2=Tucker |first2=Thomas |year=2002 |title=It's As Easy As abc |url=https://www.ams.org/notices/200210/fea-granville.pdf |journal=Notices of the AMS |volume=49 |issue=10 |pages=1224–1231}}</ref>
* The [[Fermat–Catalan conjecture]], a generalization of [[Fermat's Last Theorem]] concerning powers that are sums of powers.{{sfnp|Pomerance|2008}}
* The [[Dirichlet L-function|''L''-function]] ''L''(''s'', ''χ<sub>d</sub>'') formed with the [[Legendre symbol]], has no [[Siegel zero]], given a uniform version of the ''abc'' conjecture in [[number field]]s, not just the ''abc'' conjecture as formulated above for rational integers.{{sfnp|Granville|Stark|2000}}
* A [[polynomial]] ''P''(''x'') has only finitely many [[perfect powers]] for all [[integers]] ''x'' if ''P'' has at least three [[simple zero]]s.<ref name="Ref_a">[http://www.math.uu.nl/people/beukers/ABCpresentation.pdf The ABC-conjecture], Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.</ref>
* A generalization of [[Tijdeman's theorem]] concerning the number of solutions of ''y<sup>m</sup>'' = ''x<sup>n</sup>'' + ''k'' (Tijdeman's theorem answers the case ''k'' = 1), and Pillai's conjecture (1931) concerning the number of solutions of ''Ay<sup>m</sup>'' = ''Bx<sup>n</sup>'' + ''k''.
* As equivalent, 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 that for all coprime integers ''x'', ''y'', the radical of ''f''(''x'', ''y'') exceeds ''C'' · max{|''x''|, |''y''|}<sup>''n''−''β''</sup>.<ref>{{harvtxt|Mollin|2009}}; {{harvtxt|Mollin|2010|p=297}}</ref>
* all the polynominals (x^n-1)/(x-1) have an infinity of square-free values.<ref>{{harvtxt|Browkin|2000|p=10}}</ref>
* As equivalent, the modified [[Szpiro conjecture]], which would yield a bound of rad(''abc'')<sup>1.2+''ε''</sup>.{{sfn|Oesterlé|1988}}
* {{harvtxt|Dąbrowski|1996}} has shown that the ''abc'' conjecture implies that [[Brocard's problem|the Diophantine equation ''n''! + ''A'' = ''k''<sup>2</sup>]] has only finitely many solutions for any given integer ''A''.
* There are ~''c''<sub>''f''</sub>''N'' positive integers ''n'' ≤ ''N'' for which ''f''(''n'')/B' is square-free, with ''c''<sub>''f''</sub> > 0 a positive constant defined as:{{sfnp|Granville|1998}}{{block indent|<math>c_f = \prod_{\text{prime }p} x_i \left ( 1 - \frac{\omega\,\!_f (p)}{p^{2+q_p}} \right ).</math>}}
*The [[Beal conjecture]], a generalization of Fermat's Last Theorem proposing that if ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''A<sup>x</sup>'' + ''B<sup>y</sup>'' = ''C<sup>z</sup>'' and ''x'', ''y'', ''z'' > 2, then ''A'', ''B'', and ''C'' have a common prime factor. The ''abc'' conjecture would imply that there are only finitely many counterexamples.
*[[Néron–Tate height#Lower bounds for the Néron–Tate height|Lang's conjecture]], a lower bound for the [[height function|height]] of a non-torsion rational point of an elliptic curve.
* A negative solution to the [[Erdős–Ulam problem]] on dense sets of Euclidean points with rational distances.<ref>
{{citation
 | last1 = Pasten | first1 = Hector
 | doi = 10.1007/s00605-016-0973-2
 | issue = 1
 | journal = [[Monatshefte für Mathematik]]
 | mr = 3592123
 | pages = 99–126
 | title = Definability of Frobenius orbits and a result on rational distance sets
 | volume = 182
 | year = 2017| s2cid = 7805117
 }}
</ref>
* An effective version of [[Siegel's theorem on integral points|Siegel's theorem about integral points on algebraic curves]].<ref>{{cite journal | arxiv=math/0408168 | last1=Surroca | first1=Andrea | title=Siegel's theorem and the abc conjecture |url=https://www.rivmat.unipr.it/fulltext/2004-3s/pdf/22.pdf | date=2004 |journal= Rivista Matematica dell'Universita' di Parma, Atti del Secondo Convegno Italiano di Teoria dei Numeri |volume=3* |issue=7 |pages=323-332}}</ref>