Let $D$ be a discriminant, i.e., $D \equiv 1 \pmod{4}$ or $D \equiv 0 \pmod{4}$. Let $\mathcal{S}(D)$ be the set of positive integers for which there exists a binary quadratic form $f$ with integer coefficients and discriminant $D$ and integers $x,y$ such that $f(x,y) = n$.

In a recent preprint, Dietmann and Esholtz (https://arxiv.org/abs/1810.03203) proved that there exists a positive number $C$ such that for infinitely many $X$ such that the interval $[X, X + C\log X (\log \log |D|)^{-1})$ does not contain an element of $\mathcal{S}(D)$.

This gap result is analogous to large gap results for primes, where it is known (due to work of Maynard, Ford, Green, Konyagin, Tao) that the gap between consecutive primes $p_{n+1}, p_n$ can be as large as

$$C\frac{\log p_n \log \log p_n \log \log \log \log p_n}{\log \log \log p_n}$$

for some $C > 0$. Moreover, in the prime case there is Cramer's conjecture, which asserts that there is a positive number $c$ such that for any $X$ the interval $[X, X + c(\log X)^2]$ contains a prime.

Since the average gap between elements of $\mathcal{S}(D)$ should be about $c_D \sqrt{\log X}$ for some positive number $c_D$ depending on $D$, one can consider the analogous 'Cramer's conjecture' that the square of the average gap should be enough to cover all primes.

In particular, is there evidence (other than the comparison to the prime case) that the following conjecture holds?

There exists a positive number $c_D$ such that for any positive real number $X$ the interval $[X, X + c_d(\log X)^2]$ contains an element of $\mathcal{S}(D)$?

  • $\begingroup$ By "contains an element of $\mathcal S(D)$" in the second paragraph, do you mean "does not contain an element of $\mathcal S(D)$"? $\endgroup$ – Greg Martin Oct 9 '18 at 19:01
  • $\begingroup$ Indeed, it should read “does not contain an element”. $\endgroup$ – Rainer Dietmann Oct 9 '18 at 19:13
  • $\begingroup$ Yes, that was a typo. I have fixed it in the current version $\endgroup$ – Stanley Yao Xiao Oct 9 '18 at 19:32

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