abc conjecture

Remarks

The conjecture was formulated by Joseph Oesterlé and David Masser in 1985. Shinichi Mochizuki announced a proof in 2012; it is not accepted by most experts, and the conjecture is generally treated as open. The analogue for polynomials is a theorem, the Mason–Stothers theorem. The $\varepsilon$ cannot be removed. For each $k \ge 1$, take $a = 1$, $c = 2^{6k}$ and $b = c - 1$; since $2^6 = 64 \equiv 1 \pmod 9$, the number $9$ divides $b$, so $\operatorname{rad}(abc) \le 2 \cdot b/3 < c$. For example, $1 + 8 = 9$ has $\operatorname{rad}(1 \cdot 8 \cdot 9) = 6 < 9$, and $5 + 27 = 32$ has radical $30 < 32$. The conjecture would imply many results, for instance that the Fermat equation $x^n + y^n = z^n$ has no solutions in positive integers for all large $n$.