The abc Conjecture Revisited

2026/07/08 by Patrick Letendre
Mathematics · #math.NT

paper · pdf

Abstract

We propose a new abc-type conjecture. We motivate the conjecture and illustrate its relevance through several applications. Our main result concerns the function W(x,y) := ∑_{j = 1}^yω(x+j) (y ∈ ℕ, x ∈ ℤ_{≥ 0}) where ω(n) denotes the number of distinct prime divisors of n. The new conjecture implies that, for each fixed y ∈ ℕ, \limsup_{x → ∞} (W(x,y)loglog x)/(log x) = 1.

Related

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• A more general abc conjecture
• Applications of the abc conjecture to powerful numbers
• The abc conjecture is true almost always
• On a conjecture of R. M. Murty and V. K. Murty II