The abc conjecture, or the Oesterlé-Masser conjecture, was first proposed by Joseph Osterlé and David Masser. The conjecture states that for all ɛ > 0, there are finitely many positive integer triples which have no common factor, a, b, and c, such that a + b = c and c > d^{(1 + ɛ)}, where d is the product of the distinct prime factors of the product abc. For example:

If a = 17, b = 18 = 2 x 3 x 3, c = 17 + 18 = 35 = 5 x 7, then d = 2 x 3 x 5 x 7 x 17 = 3570. Since d is greater than c, for all ɛ > 0, c is not greater than d^{(1 + ɛ)}.

However, there are a few exceptions for a, b, and c, where c is greater than d^{(1 + ɛ)}.

In August of 2012, Shinichi Mochizuki released a serious claim to a proof, but an error in one of his articles, detected in October, put a hold on any further analysis, and the conjecture still stands as a conjecture, rather than a theorem.

The abc conjecture has been described as “the most important unsolved problem in Diophantine analysis (Goldfield 1996).” This conjecture gives a conditional proof for several theorems and conjectures, such as Fermat’s Last Theorem, but only for large exponents n. Its impossibility to prove under all conditions for a, b, c, d, and ɛ, thus far, at least, provokes deep questions in number theory.

Now here, enjoy this gif of a dancing penguin with an A.