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Representations of positive integers by three almost-prime squares

This paper improves upon previous results by proving that every sufficiently large integer $N$ satisfying specific congruence conditions can be represented as the sum of three squares where the product of the bases is a $P_{67}$ -number, each base is a $P_{27}$ -number, or a related one-dimensional variant involves a $P_{18}$ -number, achieved by combining higher-dimensional sieves, a Richert-type weighted sieve, and Bombieri-Vinogradov type estimates.

Original authors: Yue-Feng She, Yu-Chen Sun, Guang-Liang Zhou

Published 2026-06-04

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CC BY 4.0

Original authors: Yue-Feng She, Yu-Chen Sun, Guang-Liang Zhou

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 have a giant, magical number, let's call it N. You want to build this number using a very specific recipe: you need to find three other numbers, square them (multiply them by themselves), and add them up to get N.

Mathematicians have known for a long time that if N is big enough and follows a few simple rules (like not being a multiple of 5 and leaving a remainder of 3 when divided by 24), you can almost always do this. But there's a catch: usually, the three numbers you use to build N are just regular integers.

This paper is about a much harder version of that puzzle. The authors want to know: Can we build this giant number using three "almost-prime" squares?

What is an "Almost-Prime"?

Think of prime numbers (like 2, 3, 5, 7) as the "purest" building blocks of math. They can't be broken down further.

A Prime is a block with exactly one "ingredient" (itself).
An Almost-Prime is a block that is almost pure. It might have a few extra ingredients mixed in.
- A P2-number is a number with at most 2 prime ingredients (like $6 = 2 \times 3$ ).
- A P67-number is a number with at most 67 prime ingredients.

The more ingredients a number has, the "less pure" it is. The goal of this paper is to find a recipe where the ingredients are as "pure" as possible.

The Big Achievement

The authors, Yue-Feng She, Yu-Chen Sun, and Guang-Liang Zhou, managed to improve a previous record set by another mathematician named Waibel.

The Old Recipe (Waibel):
Waibel showed that you could build N using three squares where the product of the three numbers had at most 72 ingredients.

Analogy: Imagine you are building a tower. Waibel said, "You can use three bricks, as long as the total number of cracks in all three bricks combined is no more than 72."

The New Recipe (This Paper):
The authors proved you can do better. They showed you can build N in two different, even stricter ways:

The "Teamwork" Approach (Theorem 1.1):
You can find three numbers ( $x_1, x_2, x_3$ ) such that when you multiply them all together, the total number of prime ingredients is at most 67.
- Analogy: You still have three bricks, but now the total number of cracks in the whole stack is reduced to 67. It's a tighter, cleaner build.
The "Individual Purity" Approach (Theorem 1.2):
This is even more impressive. They proved that each of the three numbers individually is very pure. Each number has at most 27 prime ingredients.
- Analogy: Instead of just checking the total cracks in the stack, they proved that every single brick you use has fewer than 27 cracks. You are using three very high-quality, nearly pure bricks.

The Second Puzzle: A Different Shape

The paper also tackles a slightly different version of the problem. Instead of three squares ( $x^2 + y^2 + z^2$ ), they looked at a shape that looks like $x^2 + y^2 + (2az)^2$ .

The Old Result: A mathematician named Banerjee showed you could do this if the last number ( $z$ ) had at most 118 ingredients.
The New Result: The authors proved you can do this if $z$ $z$ has at most 18 ingredients.
- Analogy: Banerjee said, "You can build this shape if the final piece has up to 118 cracks." The authors said, "No, we can build it with a final piece that has only 18 cracks." That is a massive improvement in quality.

How Did They Do It?

They didn't just guess. They used a sophisticated mathematical toolkit called Sieve Methods.

The Sieve Analogy: Imagine you have a giant bucket of sand (all possible numbers). You want to find the gold nuggets (the "almost-prime" numbers that fit your recipe).
- First, you use a coarse sieve to remove the big rocks (numbers with too many factors).
- Then, you use a finer sieve to remove the medium rocks.
- Finally, you use a very fine, weighted sieve (a method introduced by Cai and refined by the authors) to filter out the dust, leaving only the purest gold.

They combined this "sifting" process with advanced estimates (like the "Bombieri-Vinogradov" and "Waibel" results) to prove that the gold nuggets are not just rare, but actually abundant enough to guarantee a solution for any sufficiently large number N.

Summary

In simple terms, this paper is a victory for mathematical precision. The authors took a known puzzle about building numbers from squares and proved that you can do it using ingredients that are much "purer" (have fewer prime factors) than anyone had proven possible before. They tightened the rules from "72 cracks" down to "67 cracks total" or "27 cracks per brick," and improved a related puzzle from "118 cracks" down to "18 cracks."

Technical Summary: Representations of Positive Integers by Three Almost-Prime Squares

Problem Statement
The paper addresses the representation of positive integers as sums of three squares, specifically focusing on the constraints imposed on the prime factorization of the variables. While Legendre and Gauss established that an integer $m$ is representable as $m = x^2 + y^2 + z^2$ if and only if $m \neq 4^k(8\ell+7)$ , the authors investigate representations where the variables $x_i$ are "almost-primes" (integers with a bounded number of prime factors).

The primary focus is on integers $N$ satisfying the conditions $N \equiv 3 \pmod{24}$ and $5 \nmid N$ . The paper seeks to improve upon previous results regarding the number of prime factors ( $P_r$ -numbers) required for the variables $x_1, x_2, x_3$ in the equation $N = x_1^2 + x_2^2 + x_3^2$ . Specifically, it aims to lower the bound $r$ such that $x_1x_2x_3$ is a $P_r$ -number, or such that each individual $x_i$ is a $P_r$ -number.

A secondary problem involves a related conjecture by Sun concerning representations of the form $N = x^2 + y^2 + (2az)^2$ , aiming to improve the bound on the almost-prime nature of $z$ .

Methodology
The proofs rely on a combination of analytic number theory techniques, specifically:

Higher-Dimensional Sieve: The authors employ a sieve method in three dimensions to handle the sum of three squares.
Richert-Type Weighted Sieve: They utilize a weighted sieve method introduced by Cai [6], which allows for the optimization of the number of prime factors by assigning weights to the variables based on their prime divisors.
Distribution Estimates: The core of the argument depends on the "level of distribution" for the ternary quadratic form $x_1^2 + x_2^2 + x_3^2$ . The authors utilize a Bombieri-Vinogradov type result and a diagonal norm estimate provided by Waibel [21].
Theta Series and Modular Forms: To establish the necessary error term bounds for the distribution of representations, the paper analyzes the Fourier coefficients of theta series associated with the quadratic forms. Specifically, it uses Waibel's Theorem 2 to bound the difference between the genus average and the actual representation number.
One-Dimensional Reduction: For the secondary result (Theorem 1.3), the method is adapted to a one-dimensional sieve setting, applying similar weighted sieve arguments to a single variable $z$ in the form $N = x^2 + y^2 + (2az)^2$ .

Key Contributions and Results

Theorem 1.1: For sufficiently large integers $N$ satisfying $N \equiv 3 \pmod{24}$ and $5 \nmid N$ , the authors prove that $N$ can be written as $N = x_1^2 + x_2^2 + x_3^2$ where the product $x_1x_2x_3$ is a $P_{67}$ -number (an integer with at most 67 prime factors). The number of such representations is shown to be $\gg N^{1/2-\epsilon}$ .
- Improvement: This improves upon Waibel [21], who obtained a $P_{72}$ -number.
Theorem 1.2: Under the same conditions, the authors prove that $N$ can be represented as $N = x_1^2 + x_2^2 + x_3^2$ where each individual variable $x_i$ is a $P_{27}$ -number. The number of such representations is also $\gg N^{1/2-\epsilon}$ .
- Note: This result is derived from the same weighted-sieve argument as Theorem 1.1 but applies the almost-prime estimate to each coordinate separately.
Theorem 1.3: Addressing a related problem, the authors prove that every sufficiently large integer $m$ not of the form $4^k(8\ell+7)$ can be written as $m = x^2 + y^2 + (2az)^2$ , where $x, y, a$ are non-negative integers and $z$ is a $P_{18}$ -number. The number of such representations is $\gg m^{1/2-\epsilon}$ .
- Improvement: This significantly improves upon a result by Banerjee [1], who obtained a $P_{118}$ -number.
Corollary 1.1: As a direct consequence of Theorem 1.3, the paper shows that sufficiently large integers can be represented in the form $m = x^2 + y^2 + 2^{2a} + (2bz)^2$ with $z$ being a $P_{18}$ -number.

Significance
The paper claims to improve the state-of-the-art bounds for representing integers as sums of squares of almost-primes. By refining the weighted sieve parameters and leveraging Waibel's distribution estimates, the authors successfully reduce the required number of prime factors for the variables in these representations. The reduction from $P_{72}$ to $P_{67}$ (and $P_{118}$ to $P_{18}$ in the related case) demonstrates the effectiveness of combining higher-dimensional sieves with modern estimates on the distribution of quadratic forms. The work contributes to the broader conjecture that such integers can be represented as sums of squares of actual primes, by narrowing the gap between "almost-primes" and primes.