The abc Conjecture

Establish the abc Conjecture: For every ε > 0 there exists a constant C_ε > 0 such that for all pairwise coprime nonzero integers a, b, c with a + b = c, one has |c| ≤ C_ε · rad(abc)^{1+ε}, where rad(abc) denotes the product of the distinct prime factors of abc.

Background

The abc Conjecture of Masser–Oesterlé is assumed in this paper to derive strong bounds on the number of prime values taken by Ramanujan’s tau-function. Under this conjecture, the authors prove that τ misses almost all primes, giving S(X) = O(X{9/10} log X).

The conjecture serves as a central conditional hypothesis enabling the main analytic estimates (the ‘engine’) concerning integer points near certain hyperelliptic curves, which in turn control the occurrence of prime values of τ.

References

To study \mathcal P\cap \mathcal V, the odd primes (up to sign) that occur as \tau-values, we employ the celebrated $abc$ Conjecture of Masser and Oesterl e Exp.~694.

— ABC implies that Ramanujan's tau function misses almost all primes  (2603.29970 - Angdinata et al., 31 Mar 2026) in Section 1 (Introduction and statement of results), Conjecture (abc Conjecture)