Longer Gaps Between Values of Binary Quadratic Forms

Abstract

We prove new lower bounds on large gaps between integers that are sums of two squares or are represented by any binary quadratic form of discriminant $D$, improving the results of Richards. 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 $D$, then $$ \begin{align*} & \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log s_n} \gg \frac{|D|}{\varphi(|D|)\log |D|}, \end{align*}$$improving a lower bound of $\frac {1}{|D|}$ of Richards. In the special case of sums of two squares, we improve Richards’s bound of $1/4$ to $\frac {390}{449}=0.868\ldots $. We also generalize Richards’s result in another direction: if $d$ is composite we show that there exist constants $C_d$ such that for all integer values of $x$ none of the values $p_d(x)=C_d+x^d$ is a sum of two squares. Let $d$ be a prime. For all $k\in {\mathbb {N}}$, there exists a smallest positive integer $y_k$ such that none of the integers $y_k+j^d, 1\leq j \leq k$, is a sum of two squares. Moreover, $$ \begin{align*} & \limsup_{k \rightarrow \infty} \frac{k}{\log y_k} \gg \frac{1}{ \sqrt{\log d}}. \end{align*}$$

A polynomial analogue of Jacobsthal function

“…Our study of j f is motivated by Theorem 5 in [4]. This result gives an upper bound for the least integer γ k > 0 such that all the numbers γ k + j d for 1 ⩽ j ⩽ k are not sums of two squares.…”

Gaps in a Sumset of a Polynomial With Itself

“…By increasing γ 1 to γ, if necessary, we conclude that (X, X + γX (1−1/d) 2 ) contains an element of S P for every X ≥ 1. Therefore, for every j ∈ N choosing X = s j , we deduce (5).…”

“…For P (x) = x 2 the lower bound of the form s j+1 − s j ≫ log s j is due to Richards [11]. Then, the constant implicit in ≫ has been improved by Dietmann and Elsholtz [5], and by Kalmynin and Konyagin [9] (both preprints were included in a subsequent paper [6]). The upper bound s j+1 − s j ≪ s 1/4 j is due to Bambah and Chowla [1], with subsequent improvements by Uchiyama [14], Shiu [12], [13], and Jameson [8].…”