A note on Ono's numbers associated to imaginary quadratic fields (Q5938551)

For an imaginary quadratic field \(k_D\) with discriminant \(-D\). T. Ono defined Ono's number \(p_D\) and showed some relations between \(p_D\) and the class number \(h_D\) of \(k_D\). Now it is well-known that (1) \(p_D=1\) if and only if \(h_D=1\), (2) \(p_D=2\) if and only if \(h_D=2\), (3) \(p_D\leq h_D\), (4) \(h_D\leq 2^{p_D}\) holds for \(D\) whose square-free part is less than or equal to 8173 [cf. \textit{T. Ono}, Arithmetic of algebraic groups and its applications, St. Paul's international exchange series occasional papers VI, St. Paul's Univ., Tokyo (1986)]. NEWLINENEWLINENEWLINEIn this paper, the authors first give an upper bound for \(p_D\), and next by using the upper bound show that there exist infinitely many \(D\) such that \(h_D\leq 2^{p_D}\) does not hold. They also note that \(h_D> 2^{p_D}\) appears first at \(D= 37123\), \(h_{37123}= 17\), \(p_{37123}= 4\).