abc conjecture
Number theory
For coprime positive integers with a + b = c, the conjecture that for every ε > 0 only finitely many such triples satisfy c > rad(abc)^(1+ε), where rad denotes the product of the distinct primes dividing abc.
What is usually left out
Two conditions are routinely dropped and both are essential. Without coprimality the statement is false immediately. And the bound involves the radical, which discards exponents entirely — that is the whole content, since it says a sum of numbers with few distinct prime factors cannot itself have many repeated ones. Mochizuki claimed a proof in 2012; it has not achieved consensus acceptance.
Related terms
In the atlas
- abc conjecture — Number theory, open