Abc conjecture


The abc conjecture is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ' and ' that are relatively prime and satisfy '. The conjecture essentially states that the product of the distinct prime factors of ' cannot often be much smaller than . A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".
The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture.
Various attempts to prove the abc conjecture have been made, but none have gained broad acceptance. Shinichi Mochizuki claimed to have a proof in 2012, but the conjecture is still regarded as unproven by the mainstream mathematical community.

Formulations

Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integer ', the radical of ', denoted ', is the product of the distinct prime factors of '. For example,
'
'
If a, b, and c are coprime positive integers such that a + b = c, it turns out that "usually" . The abc conjecture deals with the exceptions. Specifically, it states that:
An equivalent formulation is:
Equivalently :
A fourth equivalent formulation of the conjecture involves the quality ''q of the triple, which is defined as
For example:
A typical triple of coprime positive integers with a'' + b = c will have c < rad, i.e. q < 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 fourth formulation is:
Whereas it is known that there are infinitely many triples of coprime positive integers with a + b = c such that q > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or even q > 1.0001, etc. In particular, if the conjecture is true, then there must exist a triple that achieves the maximal possible quality q.

Examples of triples with small radical

The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with c > rad. For example, let
The integer b is divisible by 9:
Using this fact, the following calculation is made:
By replacing the exponent 6n with other exponents forcing b to have larger square factors, the ratio between the radical and c can be made arbitrarily small. Specifically, let p > 2 be a prime and consider
Now it may be plausibly claimed that b is divisible by p2:
The last step uses the fact that p2 divides 2p − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2p = p2 + 1.
And now with a similar calculation as [|above], the following results:
A list of the [|highest-quality triples] is given below; the highest quality, 1.6299, was found by Eric Reyssat for

Some consequences

The abc conjecture has a large number of consequences. These include both known results and conjectures for which it gives a conditional proof. The consequences include:
  • Roth's theorem on Diophantine approximation of algebraic numbers.
  • The Mordell conjecture.
  • As equivalent, Vojta's conjecture in dimension 1.
  • The Erdős–Woods conjecture allowing for a finite number of counterexamples.
  • The existence of infinitely many non-Wieferich primes in every base b > 1.
  • The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers.
  • Fermat's Last Theorem has a famously difficult proof by Andrew Wiles. However it follows easily, at least for, from an effective form of a weak version of the abc conjecture. The abc conjecture says the lim sup of the set of all qualities 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.
  • The Fermat–Catalan conjecture, a generalization of Fermat's Last Theorem concerning powers that are sums of powers.
  • The L-function L formed with the Legendre symbol, has no Siegel zero, given a uniform version of the abc conjecture in number fields, not just the abc conjecture as formulated above for rational integers.
  • A polynomial P has only finitely many perfect powers for all integers x if P has at least three simple zeros.
  • A generalization of Tijdeman's theorem concerning the number of solutions of ym = xn + k, and Pillai's conjecture concerning the number of solutions of Aym = Bxn + 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 such that for all coprime integers x, y, the radical of f exceeds C · maxn−''β.
  • All of the polynomials / have infinitely many square-free values..
  • As equivalent, the modified Szpiro conjecture, which would yield a bound of rad1.2+ε''.
  • 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/B' is square-free, with cf > 0 a positive constant defined as:
  • The Beal conjecture, a generalization of Fermat's Last Theorem proposing that if A, B, C, x, y, and z are positive integers with Ax + By = Cz 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.
  • Lang's conjecture, a lower bound for the 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.
  • An effective version of Siegel's theorem about integral points on algebraic curves.

    Theoretical results

The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. Bounds are known that are exponential. Specifically, the following bounds have been proven:
In these bounds, K1 and K3 are constants that do not depend on a, b, or c, and K2 is a constant that depends on ε but not on a, b, or c. The bounds apply to any triple for which c > 2.
There are also theoretical results that provide a lower bound on the best possible form of the abc conjecture. In particular, showed that there are infinitely many triples of coprime integers with a + b = c and
for all k < 4. The constant k was improved to k = 6.068 by.

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 < 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.
scope="col" q > 1q > 1.05q > 1.1q > 1.2q > 1.3q > 1.4
c < 102644200
c < 103311714831
c < 10412074502283
c < 10541824015251136
c < 1061,2686673791022911
c < 1073,4991,6698562106017
c < 1088,9873,8691,8013849825
c < 10922,3168,7423,69370614434
c < 101051,67718,2337,0351,15921851
c < 1011116,97837,61213,2661,94732764
c < 1012252,85673,71423,7733,02845574
c < 1013528,275139,76241,4384,51959984
c < 10141,075,319258,16870,0476,66576998
c < 10152,131,671463,446115,0419,497998112
c < 10164,119,410812,499184,72713,1181,232126
c < 10177,801,3341,396,909290,96517,8901,530143
c < 101814,482,0652,352,105449,19424,0131,843160

As of May 2014, ABC@Home had found 23.8 million triples.
RankqabcDiscovered by
11.62992310·109235Eric Reyssat
21.626011232·56·73221·23Benne de Weger
31.623519·13077·292·31828·322·54Jerzy Browkin, Juliusz Brzezinski
41.5808283511·13228·38·173Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
51.567912·3754·7Benne de Weger

Note: the quality ''q'' of the triple is defined above.