Longer gaps between values of binary quadratic forms

Let $s_1, s_2, \ldots$ be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant $D$. We show that $\limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log s_n} \ge \frac{φ(|D|)}{2|D|(1+\log φ(|D|))}\gg \frac{1}{\log \log |D|},$ improving a lower bound of $\frac{1}{|D|}$ of Richards (1982). In the special case of sums of two squares, we improve Richards's bound of $1/4$ to $\frac{195}{449}=0.434\ldots$. We also generalize Richards's result in another direction and establish a lower bound on long gaps between sums of two squares in certain sparse sequences.

No reviews yet.

No comments yet.