A Short Note on the Divisibility of Class Numbers of Real Quadratic Fields

For any integer l ≥ 1, let p1, p2, . . . , p_l+2 be distinct prime numbers ≥ 5. For all real numbers X > 1, we let N3,l (X) denote the number of real quadratic fields K whose absolute discriminant dK ≤ X and dK is divisible by (p1 . . . p_l+2) together with the class number hK of K divisible by 2^l · 3. Then, in this short note, by following the method in [3], we prove that N3,l (X) ≫ X ^7/8 for all large enough X’s.