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

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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

 .

For example:

 
 

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

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

 

which has a quality equal to

 

Relation to Fermat's Last Theorem

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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

 

Then this defines a triple whose quality is by definition:

 

But since   is divisible by the same prime numbers as xyz, this is the same as

 

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

 

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

 

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

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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:
  • conjectured:


See also

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References

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