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Longer gaps between values of binary quadratic forms. (arXiv:1810.03203v1 [math.NT])
来源于:arXiv
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 \[
\limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log s_n}
\ge \frac{\varphi(|D|)}{2|D|(1+\log \varphi(|D|))}\gg \frac{1}{\log \log
|D|}, \] 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. 查看全文>>