\textit{D. Miklaszewski} [Bull. Belg. Math. Soc. 3, No. 2, 239-242 (1996; Zbl 0848.34028)] found that the differential equation \[ \frac{dz}{dt}=z^2+r e^{i t} \] has no periodic solutions for some choice of the parameter \(r\) provided that the following conjecture is true. There is some integer \(N\) such that the elements of the sequence defined by \(a_1=1\) and \(a_n=(1/n)\sum_{k=1}^{n-1}a_k a_{n-k}\) satisfy the inequalities \(a_n^2N\). This conjecture is proved. Also, it is shown that there is a strictly increasing sequence of real numbers \(\{r_j\}_{j=1}^\infty\) with \(\lim_{j\to\infty} r_j=\infty\) such that if \(r\neq r_j\), then the differential equation has exactly one periodic solution, and if \(r=r_j\), then there are no periodic solutions. Moreover, the numbers \(2 \sqrt{r_j}\) are zeros of the Bessel function \[ J_0(s)=\sum_{m=0}^\infty \frac{(-s^2/4)^m}{(m!)^2}. \] Finally, an abstract generalization of these results is obtained for differential equations of the type \[ \frac{dz}{dt}=z^n+x(1+q_1(x) z+\cdots+q_{n-1}(x)z^{n-1}), \] where \(x=re^{it}\) and the \(q_j\) are polynomials whose degrees are uniformly bounded. In particular, it is proved that there are systems of this type with \(n\geq 3\) and no periodic solutions.