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Primes of the form $p^2 + nq^2$

This paper proves that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime for $n \equiv 0, 4 \pmod 6$ , establishing an asymptotic count and verifying the Friedlander–Iwaniec "Gaussian primes conjecture" for $n=4$ by applying Type I/II sum methods in $\mathbf{Q}(\sqrt{-n})$ enhanced with recent advances in Gowers norms and concatenation theorems.

Original authors: Ben Green, Mehtaab Sawhney

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ben Green, Mehtaab Sawhney

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master chef trying to bake a very specific kind of cake. The recipe is simple: take two ingredients, let's call them Prime A and Prime B. Mix them together in a special mathematical bowl using the formula $A^2 + n \times B^2$ . Your goal is to find out if the resulting mixture is also a Prime Number.

For most numbers $n$ , this is a guessing game. But this paper, written by Ben Green and Mehtaab Sawhney, proves that if you choose $n$ to be a number that leaves a remainder of 0 or 4 when divided by 6 (like 0, 4, 6, 10, etc.), you will never run out of these special cakes. In fact, you can bake an infinite number of them.

Here is how they solved the puzzle, explained through everyday analogies:

1. The Problem: Finding the "Golden Triplets"

In the world of numbers, Prime Numbers are the building blocks (like 2, 3, 5, 7, 11). They are special because they can only be divided by 1 and themselves.

The authors are looking for a "Golden Triplet":

A prime number ( $p$ ).
Another prime number ( $q$ ).
A third number formed by $p^2 + nq^2$ , which must also be prime.

Think of it like a lock with three tumblers. You need to find a combination where all three tumblers click into place at the same time. For a long time, mathematicians knew you could find pairs, but proving you could find three in this specific arrangement was a massive challenge.

2. The Strategy: The "Sieve" and the "Map"

To find these numbers, the authors didn't just check numbers one by one (that would take forever). Instead, they used a two-step strategy:

Step A: The Sieve (The Colander)
Imagine you have a giant colander (a sieve) full of numbers. You want to filter out everything that isn't a prime.

Type I Sums: This is like checking the colander for big clumps of dirt. The authors proved they could filter out the obvious non-primes efficiently.
Type II Sums: This is the tricky part. It's like looking for tiny, hidden grains of sand that look like dirt but aren't. This is where the math gets very hard. The authors had to prove that even these tiny, hidden patterns don't mess up their count.

Step B: The Map (The Number Field)
To make the math easier, the authors didn't just look at regular numbers on a straight line. They imagined a map of a different world called a Number Field (specifically, a world involving $\sqrt{-n}$ ).

Think of this as switching from a flat 2D map to a 3D globe. Sometimes, a problem that looks impossible on a flat map becomes easy when you view it from a different angle. By moving their problem into this "imaginary" world, they could use powerful tools to count the primes more accurately.

3. The Secret Weapon: "Gowers Norms"

The real magic in this paper lies in how they handled the "Type II" sums (the hidden grains of sand). They used a tool from a different branch of math called Additive Combinatorics.

Imagine you are trying to detect if a song is being played in a noisy room.

If the song is just random noise, it's hard to hear.
If the song has a strong, repeating rhythm (a pattern), you can hear it even through the noise.

The authors used Gowers Norms to measure the "rhythm" of the numbers.

They proved that if the numbers don't have a strong, predictable rhythm (which primes generally don't), then the "noise" cancels itself out.
This allowed them to ignore the messy parts of the equation and focus only on the clean, prime parts. They used very recent, cutting-edge discoveries about these "rhythms" (called concatenation theorems) to make their proof work.

4. The Result: A Counting Machine

Once they filtered out the noise and mapped the problem correctly, they could finally count the cakes.

They didn't just say "there are infinite." They gave a precise asymptotic formula.

Analogy: If you ask, "How many stars are in the sky?" a simple answer is "a lot." A precise answer is "If you look at a patch of sky this big, you will see roughly $X$ stars, give or take a small margin of error."
The authors provided the exact formula for $X$ . They calculated exactly how many of these "Golden Triplets" exist up to any given size, with a very small margin of error.

5. The Special Case: The "Gaussian Primes"

The paper highlights a specific case where $n = 4$ .

In this case, the formula becomes $p^2 + 4q^2$ .
This solves a famous conjecture (the "Gaussian Primes Conjecture") that had been sitting on the shelf for decades. It confirms that you can find infinite primes of this form where both $p$ and $q$ are prime numbers.

Summary

In short, Green and Sawhney built a mathematical machine that:

Translated a difficult number problem into a different "world" (Number Fields).
Used a high-tech sieve to filter out non-primes.
Used advanced "rhythm detectors" (Gowers Norms) to ensure no hidden patterns were messing up the count.
Proved that for specific types of numbers, you can find an infinite number of these special prime combinations, and they even gave you a recipe to count exactly how many there are.

They didn't just find one; they proved the supply is endless and gave us the tools to count them all.

Technical Summary: Primes of the Form $p^2 + nq^2$

Problem Statement
The paper addresses the problem of establishing an asymptotic count for pairs of primes $(p, q)$ such that $p^2 + nq^2$ is also prime. Specifically, the authors prove that for any positive integer $n$ satisfying $n \equiv 0$ or $n \equiv 4 \pmod 6$ , there are infinitely many such primes. This resolves a specific case of the "Gaussian primes conjecture" proposed by Friedlander and Iwaniec (for $n=4$ ) and extends the study of primes represented by binary quadratic forms where the variables themselves are restricted to be prime.

The main result, Theorem 1.1, provides an asymptotic formula for the weighted sum:
$\sum_{x,y \in \mathbb{Z}} \Lambda(x)\Lambda(y)\Lambda(x^2 + ny^2) W\left(\frac{x}{N}, \frac{y}{N}\right) = \kappa_n N^2 \left( \int_{\mathbb{R}^2} W \right) + O\left( \frac{N^2 (\log \log N)^2}{\log N} \right)$
where $\Lambda$ is the von Mangoldt function, $W$ is a smooth weight function, and $\kappa_n$ is a singular series constant derived from local densities.

Methodology
The proof employs a sieve-theoretic framework adapted to the number field $K = \mathbb{Q}(\sqrt{-n})$ . The argument proceeds through several distinct stages:

Sieve Reduction: The problem is reduced to estimating sums over prime ideals in $K$ using the Duke–Friedlander–Iwaniec sieve. This requires establishing "Type I" and "Type II" estimates for weight functions $w$ defined on ideals.
- Type I sums involve sums over divisors of a fixed level $L$ .
- Type II sums are bilinear forms involving coefficients that are not necessarily structured.
  The authors utilize a dimension-3 upper bound sieve to handle the specific constraints of the problem, particularly when dealing with the von Mangoldt function which is not 1-bounded.
Gowers Norm Control: The core innovation lies in demonstrating that the required Type I and Type II estimates follow from smallness conditions on the Gowers norms of the underlying functions. The authors decompose the von Mangoldt function $\Lambda$ into a "main term" (the Cramér model $\Lambda_{\text{Cramér}}$ ) and an error term. They rely on recent quantitative results showing that the error term has small Gowers norms.
Concatenation Theorems: To handle the Type II sums, the authors employ advanced tools from additive combinatorics, specifically concatenation theorems for Gowers norms.
- They utilize recent quantitative versions of these theorems due to Kuca and to Kuca–Kravitz–Leng.
- These theorems allow the conversion of control over "Gowers–Peluse norms" (generalized norms involving probability measures) into control over standard Gowers $U^k$ norms.
- This machinery is essential for decoupling the bilinear sums and proving that if the Gowers norms of the input functions are small, the Type II sums are also small.
Analytic Number Theory: The main term is evaluated using the Prime Ideal Theorem for Hecke Größencharaktere (Proposition 8.3). The authors carefully handle the singular series $\kappa_n$ by expanding the Cramér model via inclusion-exclusion and analyzing the contribution of principal and non-principal characters, ensuring the convergence of the infinite product defining $\kappa_n$ .

Key Contributions

Resolution of the Gaussian Primes Conjecture ( $n=4$ ): The paper verifies the conjecture that there are infinitely many primes of the form $p^2 + 4q^2$ with $p, q$ prime, providing the first asymptotic formula for this specific case.
Generalization to $n \equiv 0, 4 \pmod 6$ : The result is extended to a broader class of integers $n$ , covering the parity constraints necessary for $p^2 + nq^2$ to be prime when $p, q$ are prime.
Application of Quantitative Concatenation: The paper represents a significant application of very recent developments in additive combinatorics (specifically the quasipolynomial inverse theorem and quantitative concatenation theorems) to a classical problem in analytic number theory. It bridges the gap between the qualitative understanding of Gowers norms and the quantitative bounds required for sieve methods.
Handling Non-Principal Ideals: The argument is formulated in the language of ideals in $\mathbb{Q}(\sqrt{-n})$ , successfully navigating the lack of unique factorization in general imaginary quadratic fields by working with ideal classes and fractional ideal representatives.

Results
The primary result is the asymptotic count given in Theorem 1.1. The error term is of the order $O(N^2 (\log \log N)^2 / \log N)$ , which represents a saving of a power of $\log N$ over the trivial bound. The constant $\kappa_n$ is explicitly defined as a product over primes involving Legendre symbols, reflecting the local obstructions to the form $p^2 + nq^2$ being prime.

Significance and Claims
The authors position this work as a verification of a long-standing conjecture by Friedlander and Iwaniec and a natural extension of their earlier work on primes of the form $x^2 + y^2$ . The paper claims that the main technical novelty is the treatment of Type II sums via the aforementioned concatenation theorems, which allows for the necessary savings in the sieve.

The authors note that while the argument could theoretically be modified for arbitrary positive definite binary quadratic forms, they restrict the scope to $x^2 + ny^2$ to avoid excessive notation. They also highlight that obtaining Type I information at the full level $X^{1/2}$ without absolute values remains an open question, though their current range is sufficient for the Duke–Friedlander–Iwaniec sieve. The work does not claim to solve the general problem of primes in polynomial sequences but specifically targets the "thin set" of primes where both coordinates are prime.