User:Lenthe/abc conjecture (draft)

The abc conjecture in number theory is a strong conjecture predicting restrictions on the prime factorisation of numbers a, b and c that satisfy the equation a+b=c. The conjecture implies the correctness of Fermat's Last Theorem up to finitely many counterexamples. It was first formulated by Joseph Oesterlé and David Masser in 1985. It is still unproved as of 2007.

Statement of the conjecture

The radical of an integer n, denoted rad(n), is defined to be the product of its distinct prime divisors. For example, 56 is divisible by the primes 2, and 7, so rad(56)=2x7=14.

Consider a triple (a, b, c) of coprime positive integers such that a+b=c. The quality of such a triple of integers is defined, using the log function, as

$Q(a,b,c)={\frac {\log(c)}{\log(\operatorname {rad} (abc))}}$.

For example:

$Q(2,3,5)={\frac {\log(5)}{\log(\operatorname {rad} (2\cdot 3\cdot 5))}}={\frac {\log(5)}{\log(30)}}\approx 0.47320$

$Q(3,125,128)={\frac {\log(128)}{\log(\operatorname {rad} (3\cdot 125\cdot 128))}}={\frac {\log(128)}{\log(30)}}\approx 1.42657$

The abc conjecture states that for every real number q>1 there are only finitely many coprime triples (a, b, c) with a + b = c whose quality is larger than q.

It is known that the analogous statement with q=1 is false.

The conjecture implies that there exists a triple with the highest quality. This weaker statement (the existence of a "best" triple) is sometimes referred to as the weak abc conjecture.

Best known triple

Computers have been used to compile lists of abc-triples with high quality. The triple currently holding the record for the highest quality is

$\left(2,6436341,6436343\right)=(2,3^{10}\cdot 109,23^{5})$

which has a quality equal to

${\frac {\log(6436343)}{\log(2\cdot 3\cdot 23\cdot 109)}}\approx 1.62991$

Relation to Fermat's Last Theorem

That the abc-conjecture implies that there can only be finitely many counter-examples to Fermat's Last Theorem can be seen as follows.

Assume that there exists an n>3 and coprime positive integers x, y, and z such that

$x^{n}+y^{n}=z^{n}$

Then this defines a triple whose quality is by definition:

$Q(x^{n},y^{n},z^{n})={\frac {\log(z^{n})}{\log(\operatorname {rad} (x^{n}y^{n}z^{n}))}}$

But since $x^{n}y^{n}z^{n}$ is divisible by the same prime numbers as xyz, this is the same as

$Q(x^{n},y^{n},z^{n})={\frac {\log(z^{n})}{\log(\operatorname {rad} (xyz))}}$

and since the radical of the number xyz is at most xyz itself,

$Q(x^{n},y^{n},z^{n})\geq {\frac {\log(z^{n})}{\log(xyz)}}.$

Now z is larger than x and y so log(xyz) is smaller than log(z^3), thus:

$Q(x^{n},y^{n},z^{n})>{\frac {\log(z^{n})}{\log(z^{3})}}={\frac {n\log(z)}{3\log(z)}}={\frac {n}{3}}$

Since n is at least 4, it thus follows that (x^n,y^n,z^n) is an abc-triple of quality greater than 4/3, and the abc-conjecture implies that there can only be finitely many such triples.

Some other consequences

Besides Fermat's Last Theorem up to a finite number of exceptions, a number of known and conjectured results in number theory are known to be implied by the abc conjecture.

These include:

• known:

• Roth's theorem
• The Mordell conjecture (proven by Gerd Faltings)

• conjectured:

• The Erdős–Woods conjecture except for a finite number of counterexamples
• The existence of infinitely many non-Wieferich primes
• The weak form of Hall's conjecture
• The set of consecutive triples of powerful numbers is finite
• The L function L(s,(−d/.)) formed with the Legendre symbol, has no Siegel zero
• P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.
• A generalization of Tijdeman's Theorem