Proof that Bombieri's question about the abc-conjecture is Yes

Constantin M. Petridi (1)

1 NKUA - National and Kapodistrian University of Athens

In [1] Enrico Bombieri, asked the question "For example, is it always true that c ≤ p/abc p 2 ?." in present paper we prove that the answer is: Yes. 1 The series ∞ n=1 R(n) t n s We introduce the series ∞ n=1 R(n) t n s (1) where R(n) := the multiplicative radical of n, t := a real positive variable, s := a real variable, such that s > t + 1. ∞ n=1 R(n) t n s converges because it is majorized by ∞ n=1 1 n s−t , since R(n) t ≤ n s and the latter series converges since s > t + 1. We denote the set {t > 0, s > t + 1}, the convergence region of ∞ n=1 R(n) t n s , by CR(s, t), indicating its dependance on s and t.

Domaines: Théorie des nombres [math.NT]

Constantin M. Petridi. Proof that Bombieri's question about the abc-conjecture is Yes. 2019. ⟨hal-02061466⟩