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→Working Definition: new section
That is wrong. --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 16:14, 23 October 2015 (EDT)
That is wrong. --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 16:14, 23 October 2015 (EDT)
== Working Definition ==
The ''abc-conjection'' doesn't deal with single cases, as it is a formula which holds for "almost all" triples of numbers with certain characteristics:
Let ε > 0 be fixed. Then for almost all triplets of co-prime integers '''a''', '''b''' and '''c''' with '''a+b=c''' holds:
<math>rad(a b c)^{1+\epsilon} > c</math>
Here rad(a b c) is the "radical" of the product <math>a \times b \times c</math>, i.e., the product of all unique primes dividing <math>a \times b \times c</math> - or, in other words, the square-free-part.
"Almost all triplets" means that there is only a finite number of triplets which violates the condition. Some observations:
*ε can be chosen as small as you like - as long it is greater than zero.
*the finite number of exceptions obviously becomes smaller, the bigger ε is chosen.
*There are infinitely many triplets violating <math>rad(a b c) > c</math> - that's why <math>\epsilon = 0</math> isn't allowed.