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Primes represented by shifted quadratic forms: on primitivity and congruence classes

Elena Fuchs, Catherine Hsu, James Rickards, Damaris Schindler, Katherine E. Stange

TL;DR

This work extends Iwaniec’s sieve framework for primes represented by shifted binary quadratic forms by incorporating primitivity and congruence-class restrictions. The authors develop a two-tier approach: (i) a squarefree and half-dimensional sieve to count primes $p\le N$ with $p$ represented primitively by a genus, and (ii) a Bourgain–Fuchs argument to transfer genus-level representations to a single form, yielding a lower bound of $\gg N/( log N)^{3/2}$. They also carefully analyze local representation conditions, correct and verify the associated tables, and apply the results to problems in Apollonian circle packings, providing tools to understand prime components in those packings. The methods combine refined local criteria with advanced sieve techniques to produce robust lower bounds under primitivity and congruence constraints. The results offer a concrete tool for studying primes in geometric configurations (Apollonian packings) through the arithmetic of shifted quadratic forms.

Abstract

We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an arithmetic progression. This extends the sieve methods of Iwaniec, who showed such lower bounds without the primitivity and congruence conditions. Imposing primitivity adds some subtleties to the local criteria for representation of a shifted prime: for example, some shifted quadratic forms of discriminant $5 \pmod{8}$ do not primitively represent infinitely many primes. We also provide a careful list of the local conditions under which a genus of an integral binary quadratic form represents an integer, verified by computer, and correcting some minor errors in previous statements. The motivation for this work is as a tool for the study of prime components in Apollonian circle packings [FFH+24]

Primes represented by shifted quadratic forms: on primitivity and congruence classes

TL;DR

This work extends Iwaniec’s sieve framework for primes represented by shifted binary quadratic forms by incorporating primitivity and congruence-class restrictions. The authors develop a two-tier approach: (i) a squarefree and half-dimensional sieve to count primes with represented primitively by a genus, and (ii) a Bourgain–Fuchs argument to transfer genus-level representations to a single form, yielding a lower bound of . They also carefully analyze local representation conditions, correct and verify the associated tables, and apply the results to problems in Apollonian circle packings, providing tools to understand prime components in those packings. The methods combine refined local criteria with advanced sieve techniques to produce robust lower bounds under primitivity and congruence constraints. The results offer a concrete tool for studying primes in geometric configurations (Apollonian packings) through the arithmetic of shifted quadratic forms.

Abstract

We prove lower bounds of the form for the number of primes up to primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an arithmetic progression. This extends the sieve methods of Iwaniec, who showed such lower bounds without the primitivity and congruence conditions. Imposing primitivity adds some subtleties to the local criteria for representation of a shifted prime: for example, some shifted quadratic forms of discriminant do not primitively represent infinitely many primes. We also provide a careful list of the local conditions under which a genus of an integral binary quadratic form represents an integer, verified by computer, and correcting some minor errors in previous statements. The motivation for this work is as a tool for the study of prime components in Apollonian circle packings [FFH+24]
Paper Structure (14 sections, 13 theorems, 52 equations, 1 figure, 2 tables)

This paper contains 14 sections, 13 theorems, 52 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Local conditions for representation by a genus
  4. Application to Apollonian packings.
  5. Paper structure
  6. Representability by the genus
  7. Proof of Theorem \ref{['thm:maintheorem']}
  8. Set-up
  9. The squarefree sieve
  10. Setting up a $1/2$-dimensional sieve
  11. Level of distribution
  12. Applying the sieve
  13. Passing from the genus to one form
  14. Local conditions

Key Result

Theorem 1.1

Let $f=ax^2+bxy+cy^2$ be a primitive integral binary quadratic form with discriminant $D$ not a perfect square, positive definite if $D<0$, and $(a,2D)=1$. Let $A$ be a non-zero integer and $B\in \mathbb{N}$ with ${\operatorname{gcd}}(A,B)=1$. Denote by $b_{f,A}(n)$ the characteristic function for w The notation above is taken from Iwaniec72shiftedprimes, where $D=b^2-4ac=\pm 2^{\vartheta_2}p_1^{\

Figures (1)

  • Figure 1: The construction of an Apollonian packing. Begin with four mutually tangent circles. For each triple of mutually tangent circles, a result of Apollonius asserts the existence of exactly two which complete the triple to a mutually tangent quadruple. Add these circles in. For all newly created mutually tangent triples, add in their completions. Repeat, ad infinitum.

Theorems & Definitions (18)

  • Theorem 1.1
  • Proposition 1.2: Sarnak
  • Theorem 1.3
  • Theorem 2.2: Iwaniec72shiftedprimes
  • Theorem 2.4: Iwaniec72shiftedprimes
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 8 more