The number of equations c=a+b satisfying the abc-conjecture

2009/04/13 by Constantin M. Petridi, Petridi, Constantin M.
Computer Science · Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT

paper · pdf · doi:10.48550/arxiv.0904.1935

arxiv created 2009/04/13 · openalex publication_date 2009/04/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for a positive integer c and any given ε, 0<ε<1, the number N(c) of equations c=a+b, a<b, with positive coprime integers a and b, which satisfy the inequality c < R(c)^(ε)/(1+ε)R(a)^(1)/(1+ε)R(b)^(1)/(1+ε), where R(n) is the radical of n, is for c→∞ N(c)=(1-ε)(ϕ(c))/(2)+O((ϕ(c))/(2)). An analogue for the abc-conjecture inequality c<R(abc)^1+ε (without a constant factor) will also be proved.

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