Conjecture ABC

The conjecture ABC is a Conjecture in Théorie of the numbers. She was formulated for the first time by Joseph Oesterlé and David Masser in 1985. If it were checked, she would make it possible to easily show the Théorème of Fermat , inter alia.

Formulation

Either $\ epsilon > 0$ , then there exists a constant $K_ \ epsilon$ such as, for all $a, B, c$ whole relative first between them checking $a+b=c$ , one has: $max (|has|,|B|,|C|) \ K_ \ epsilon N_0 (ABC) ^ {1+ \ epsilon}$

where $N_0 (N)$ is the radical of N, i.e. the product of the prime numbers dividing N.

Analogy with the polynomials

The idea of the conjecture ABC was formed by analogy with the polynomials. A theorem ABC is indeed available for the polynomials on a body algebraically closed of null characteristic. It is also called theorem of Mason-Stothers and is formulated as follows:

For all polynomials $a, B, c$ first between them checking $a+b=c$ , there is $max (deg \ {has, B, C \}) \ the n_0 (ABC) - 1$

where $n_0 (P)$ is the number of roots distinct from P.

This theorem makes it possible to show in an easy way the theorem of Fermat for the polynomials: the equation $x^n+y^n=z^n, X, there, Z \ in \ N_*$ does not have solutions if $n \ Ge 3$ .

Temptation is then large to find an analog for the entireties, because it would make it possible to also easily show the Théorème of Fermat.

One of the principal consequences: the theorem of Fermat

In fact, the conjecture ABC would not make it possible exactly to show the theorem of Fermat, but an asymptotic version in the direction where it is shown that there exists NR such as for all $n \ Ge N$ , $x^n+y^n=z^n$ does not have any more whole solutions. This NR would be however explicit because as we will see it depend on the $C_ \ epsilon$ , explicitly given by the demonstration of the theorem ABC.

By taking an unspecified $\ epsilon$ (or 1 to fix the ideas), when $x^n+y^n=z^n$ and that they are all nonnull, one can bring back oneself so that they are first between them while dividing by the pgcd of the three, and one thus has: $|X|^n \ the max (|X|^n,|there|^n,|Z|^n) \ K_ \ N_0 epsilon ((xyz) ^n) ^ {1+ \ epsilon}$ however $N_0 ((xyz) ^n) =N_0 (xyz)$

therefore, by writing the preceding relation for $|there|^n$ and $|Z|^n$ and by multiplying them all the three, one obtains: $|xyz|^n \ K_ \ epsilon^3 N_0 (xyz) ^ {3 (1+ \ epsilon)}$ and $N_0 (xyz) \ it |xyz|$ thus $|xyz|^ {n-3 (1+ \ epsilon)}\ K_ \ epsilon^3$ gold X, there and Z are all nonnull and one easily checks that they cannot be all of absolute value equal to 1, therefore $|xyz|\ Ge 2$ . Finally, one obtains: $2^ {n-3 (1+ \ epsilon)}\ K_ \ epsilon^3$ , which provides a limiting value to N depending explicitly on $K_ \ epsilon$ .

Other consequences

The conjecture ABC would make it possible to prove other important theorems in theory of the numbers, among which:

the Theorem of Roth
the Theorem of Baker
the Theorem of Bombieri
the Theorem of Falting previously named conjecture of Mordell

See too

PGCD
ABC@home

External bonds

http://mathworld.wolfram.com/abcConjecture.html
http://www.math.unicaen.fr/~nitaj/abc.html
http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
Site of ABC@Home. Project of Calculation distributed using BOINC in order to show the conjecture ABC by finding all the triplets ABC until 10¹⁸, even more.

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