Friable values of binary forms

Antal Balog (1) , Valentin Blomer (2) , Cécile Dartyge (3) , Gérald Tenenbaum (3)

1 Alfréd Rényi Institute of Mathematics
2 Georg-August-University = Georg-August-Universität Göttingen
3 IECN - Institut Élie Cartan de Nancy

Let F ∈ Z[X, Y ] be an integral binary form of degree g ≥2, and let Ψ_F (x, y) := card{1 ≤a, b ≤x : P^+ (F (a, b))≤ y} where as usual P^+ (n) denotes the largest prime factor of n. It is proved that Ψ_F (x, y)>> x^2 for y = x^(g−2+ε) in general, and y = x^(1/ √ e+ε) if g = 3. Better results are obtained if F is reducible.

Domaines: Théorie des nombres [math.NT]

Antal Balog, Valentin Blomer, Cécile Dartyge, Gérald Tenenbaum. Friable values of binary forms. Commentarii Mathematici Helvetici, 2012, 87 (3), pp.639-667. ⟨10.4171/CMH/264⟩. ⟨hal-01280728⟩