Abc conjecture

The abc conjecture 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.

Informally, one effect of this conjecture is that adding to co-prime numbers with many divisors will generally result in a sum which has only a few big prime factors.

An example of an exception to this rule for small ε is provided by Nature is as follows:

a = 3 b = 125 = 53 c = 128 = 27 Then reg(abc) = reg(48000) = 3*5*2 = 30 and therefore reg(48000)1+ε > 128 only if ε > .4265....

An asserted proof for the abc conjecture was posted by the prominent mathematician Shinichi Mochizuki on the internet in 2012.

See also

• PolyMath explanation
• abcathome explanation