The abc conjecture is a conjecture in number theory which states that for a given ( \varepsilon > 0 ), there are only finitely many positive integer solutions of the equation ( a + b = c ) with ( a, b ) coprime such that

[ c > \textrm{rad}(abc)^{1 + \varepsilon} ]

In 2012, Shinichi Mochizuki claimed a proof of the conjecture, but his proof has failed to achieve widespread recognition among mathematicians as of 2022.

In March 2021 his proof was published in the Publications of the Research Institute for Mathematical Sciences (RIMS), a journal of which Mochizuki was the chief editor (though he recused himself from reviewing the paper).