Talk:Abc conjecture

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expand lead, maybe a section on its importance. consider making it more reader friendly--Cronholm144 23:53, 12 May 2007 (UTC)

I've read that Fermat's Last Theorem is an easy consequence of this conjecture. If I get the chance I'll see if I can find out why and add it, unless someone else wants to do so in the meantime. --ScottAlanHill 16:54, 9 March 2006 (UTC)

Since the conjecture says nothing on the constant $C ε$ , I don't see how it should rule out the existence of finitely many solutions for FLT. However, the abc-conjecture does easily imply that for n>3 there are at most finitely many solutions: If X,Y,Z are positive integers without common divisor with $X N + Y N = Z N$ , then $r a d (X N Y N Z N) = r a d (X Y Z)$ and $rad(XYZ) \le Z^3$ and therefore the abc-conjecture (if true) yields

$Z^N < C_{\varepsilon} \operatorname{rad}(X^NY^NZ^N)^{1+\epsilon} \le C_{\varepsilon} Z^{3+3\epsilon} ,$

which implies an upper bound for Z.

Joerg Winkelmann 20:22, 12 May 2006 (UTC)

1 Math-mode formulas
2 a question
3 Formulation
4 Solution
5 Unclear Wording
6 Granville-Langevin conjecture

[edit] Math-mode formulas

Is there a reason not to use math formulas here? These are the math-style formulas:

a + b = c

$a + b = c$
rad(abc),

$\operatorname{rad}\,(abc)$
rad(abc)/c

$\frac{\operatorname{rad}\,(abc)}{c}$
rad(abc)^1+ε/c

$\frac{{\operatorname{rad}\,(abc)}^{1+\epsilon}}{c}$

Vegard 14:01, 29 September 2006 (UTC)

[edit] a question

if a, b, c are positive integers, why take the absolute value |a| + |b| + |c| ? ~~

Good point. I think this is a better version of that section, but I don't actually know the conjecture. If this is good, maybe someone can put it in the article:

[edit] Formulation

For any integer n, the radical of n is defined to be the square-free product of its distinct prime factors. In other words, the product of all of the unique prime factors of n, never raising a factor to a power greater than 1. It is denoted rad(n).

The abc conjecture states that, for any ε > 0, there exists a finite K_ε such that, for all coprime positive integers a, b and c such that a + b = c,

$a+b+c<K_\epsilon\operatorname{rad}(abc)^{1+\epsilon}.$

Orthografer 17:59, 8 May 2007 (UTC)

[edit] Solution

Lucien Szpiro announced a solution to this at the Goldfeld conference. 146.96.245.119 20:42, 24 May 2007 (UTC)

[edit] Unclear Wording

In section Some consequences, the term first group in the last sentence seems unclear to me.

While the first group of these have now been proven ...'

-- Burkhard.Plache 15:48, 6 September 2007 (UTC)

[edit] Granville-Langevin conjecture

Please help improve this article or section by expanding it.
Further information might be found on the talk page or at requests for expansion. (December 2007)

"Granville-Langevin conjecture" now redirects here, and should be explained here or forked into its own article. -- Beland 21:06, 1 December 2007 (UTC)