Arithmetic progressions in binary quadratic forms and norm forms

Elsholtz, Christian; Frei, Christopher

doi:10.1112/blms.12256

Cited by 2 publications

(1 citation statement)

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“…We will state the precise statement in Theorem 10.36. Elsholtz and Frei [EF19] have studied prime numbers represented by norm forms. The novel point of our work is that we study combinatorics for the set of tuples (x 1 , x 2 , .…”

Section: Strategy For the Proof Of Theorem 101mentioning

confidence: 99%

Constellations in prime elements of number fields

Kai¹,

Mimura²,

Munemasa³

et al. 2020

Preprint

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Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form F which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers (x, y) for which F (x, y) is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.

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Section: Strategy For the Proof Of Theorem 101mentioning

confidence: 99%

Constellations in prime elements of number fields

Kai¹,

Mimura²,

Munemasa³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

Longer Gaps Between Values of Binary Quadratic Forms

Dietmann

Elsholtz

Kalmynin

et al. 2022

International Mathematics Research Notices

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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, $$ \begin{align*} & \limsup_{k \rightarrow \infty} \frac{k}{\log y_k} \gg \frac{1}{ \sqrt{\log d}}. \end{align*}$$

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Arithmetic progressions in binary quadratic forms and norm forms

Cited by 2 publications

References 9 publications

Constellations in prime elements of number fields

Constellations in prime elements of number fields

Longer Gaps Between Values of Binary Quadratic Forms

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