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Waring's problem involving D.H. Lehmer numbers

This paper proves that every sufficiently large integer, except those congruent to 15 or 16 modulo 16, can be expressed as the sum of 14 fourth powers of D.H. Lehmer numbers, while all sufficiently large integers can be represented as the sum of 16 such fourth powers.

Original authors: Yang Qu, Rong Ma

Published 2026-06-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yang Qu, Rong Ma

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, infinite box of building blocks. In the world of mathematics, there's a famous game called Waring's Problem. The goal of this game is simple: Can you build any large number using a specific number of "power blocks"?

For example, can you build the number 100 using only squares (like $1^2, 2^2, 3^2$ )? Or cubes? The mathematicians in this paper are playing a specific version of this game using fourth powers (numbers like $1^4, 2^4, 3^4$ , which are 1, 16, 81, 256, etc.).

The Special "Lehmer" Blocks

Usually, you can use any whole number to build your tower. But the authors of this paper decided to play with a very picky set of blocks called D.H. Lehmer numbers.

Think of these numbers as "VIP members" of the number world. To be a VIP (a Lehmer number), a number has to pass a strict security check involving a prime number (let's call it $p$ ):

It must be coprime to $p$ (it shares no factors with $p$ ).
If you find its "partner" number (its modular inverse) and add them together, the result must be an odd number.

It's like a club where you can only enter if your ID number and your partner's ID number add up to an odd sum. The authors wanted to see if these specific, restricted VIP blocks were still strong enough to build any large number.

The Big Discovery

The paper proves two main things about building these numbers using fourth powers of these VIP blocks:

1. The "Almost Perfect" Rule (Theorem 1)
If you have a very large number, you can almost always build it using 14 of these VIP fourth-power blocks.

The Catch: There are two exceptions. If your target number leaves a remainder of 15 or 16 when you divide it by 16, you can't build it with just 14 blocks. It's like trying to fit a square peg in a round hole; the math just doesn't line up for those specific numbers.

2. The "Safe Bet" Rule (Theorem 2)
If you are willing to use 16 blocks instead of 14, you can build any sufficiently large number, no matter what the remainder is.

Why 16? The authors noticed that the number 1 itself is a VIP block. So, if you have a "stubborn" number (one that is 15 or 16 mod 16), you can just subtract two 1s (which are $1^4$ ) from your target. This changes the number into one that can be built with 14 blocks. Add the two 1s back, and you've used 16 blocks total to build the original number.

How They Solved It

To prove this, the authors didn't just try random combinations. They used a sophisticated mathematical toolkit called the Hardy-Littlewood Circle Method.

Imagine trying to find a specific needle in a haystack.

The Circle Method is like a giant metal detector that scans the entire haystack. It separates the "easy" parts of the problem (where the numbers behave nicely) from the "hard" parts (where they are chaotic).
The authors had to show that even with their restricted "VIP" blocks, the "easy" parts of the scan were loud and clear enough to guarantee a solution, while the "hard" parts were too quiet to cause any trouble.

They also had to prove that there were enough VIP blocks available to do the job. They showed that these numbers are distributed densely enough that you never run out of them when you need to build a large tower.

The Bottom Line

This paper is a victory for the "VIP" numbers. It shows that even though D.H. Lehmer numbers are a very specific, picky subset of integers, they are powerful enough to solve a classic, difficult math problem.

With 14 blocks: You can build almost everything (except numbers that are 15 or 16 mod 16).
With 16 blocks: You can build absolutely everything.

It's a bit like discovering that even if you only have red and blue LEGO bricks (instead of all colors), you can still build almost any castle, provided you have enough of them.

Technical Summary: Waring's Problem Involving D.H. Lehmer Numbers

Problem Statement
This paper addresses a variation of the classical Waring's problem, specifically investigating the representation of sufficiently large positive integers as sums of fourth powers of D.H. Lehmer numbers. Let $p$ be an odd prime. A D.H. Lehmer number is defined as an integer $a$ such that $1 \le a \le p$ , $\gcd(a, p) = 1$ , and $a + \bar{a} \equiv 1 \pmod 2$ , where $\bar{a}$ is the unique multiplicative inverse of $a$ modulo $p$ in the range $[1, p]$ . The set of these numbers is denoted by $L(p)$ .

The authors seek to determine the number of fourth powers of elements from $L(p)$ required to represent any sufficiently large integer $N$ . This builds upon classical results by H. Davenport, who established that every sufficiently large integer is the sum of 16 fourth powers of natural numbers, and that every sufficiently large integer not congruent to 15 or 16 modulo 16 is the sum of 14 such fourth powers. The paper aims to extend these findings to the restricted set of D.H. Lehmer numbers.

Methodology
The authors employ the Hardy-Littlewood circle method, a standard analytic technique for additive number theory problems. The core of the methodology involves:

Generating Functions: The authors define exponential sums over the set $L(p)$ . Specifically, $T(\alpha) = \sum_{x \sim P, x \in L(p)} e(\alpha x^4)$ , where $P \approx (N/100)^{1/4}$ . To handle the representation of $N$ as a sum of 14 fourth powers, they introduce an auxiliary set of integers $\{u_1, \dots, u_U\}$ , which are sums of 4 fourth powers of Lehmer numbers, and define $U(\alpha) = \sum_{h=1}^U e(\alpha u_h)$ .
Integral Representation: The number of representations, $r_{14}(N)$ , is expressed as an integral over the unit interval: $r_{14}(N) = \int_0^1 T^6(\alpha) U^2(\alpha) e(-N\alpha) d\alpha$ .
Major and Minor Arcs: The integration domain is split into major arcs $\mathfrak{M}$ $M$ (near rational numbers with small denominators) and minor arcs $\mathfrak{m}$ $m$ .
- Approximation on Major Arcs: The authors approximate the exponential sum $T(\alpha)$ on the major arcs using a singular series component $T^*(\alpha, a, q)$ and an error term $H(\alpha)$ . This relies on decomposing the sum over $L(p)$ using character sums and properties of the Möbius function.
- Estimation on Minor Arcs: Using Weyl's inequality and bounds on exponential sums involving characters, the authors establish that the contribution from the minor arcs is negligible compared to the main term.
Lemmas and Bounds: The proof relies on a series of lemmas establishing bounds for:
- Character sums (Lemma 1).
- The distribution of Lehmer numbers in short intervals (Lemma 3, Lemma 5).
- The number of solutions to equations of the form $x^4 + u_h = y^4 + u_i$ (Lemma 6).
- The density of numbers representable as sums of $s$ fourth powers of Lehmer numbers (Lemma 7).
- Bounds for $T(\alpha)$ on both major and minor arcs (Lemma 8, Lemma 14).

Key Contributions and Results
The paper establishes two primary theorems regarding the representability of large integers:

Theorem 1: Every sufficiently large positive integer $N$ , provided that $N \not\equiv 15, 16 \pmod{16}$ , can be represented as the sum of 14 fourth powers of D.H. Lehmer numbers.
Theorem 2: Every sufficiently large positive integer can be represented as the sum of 16 fourth powers of D.H. Lehmer numbers.

The proof of Theorem 1 involves showing that the main term in the integral representation dominates the error terms, ensuring $r_{14}(N) > 0$ for the specified congruence classes. Theorem 2 follows immediately from Theorem 1 and the observation that 1 is a Lehmer number; thus, integers congruent to 15 or 16 modulo 16 can be reduced to the form $N-2$ (which falls into the allowable congruence classes) by subtracting two instances of $1^4$ .

Significance and Claims
The authors state that this is the first research conducted on the Waring's problem specifically involving D.H. Lehmer numbers. The significance of the work lies in:

Extending Classical Results: It successfully adapts the mature methods used for the classical Waring's problem (specifically for $k=4$ ) to a restricted set of integers defined by modular arithmetic properties.
Validation of $G(4)=16$ : The authors note that their result, which narrows the range of integers representable as sums of 16 fourth powers to the specific subset of Lehmer numbers, provides an alternative perspective validating the classical result $G(4)=16$ .
Open Problems: The paper concludes by suggesting that the splitting of $N$ into sums of powers restricted to D.H. Lehmer numbers for other cases of Waring's problem remains an open area for future investigation.

The work is supported by the Shaanxi Fundamental Science Research Project for Mathematics and Physics and the Natural Science Basic Research Project of Shaanxi Province.

The Special "Lehmer" Blocks

The Big Discovery

How They Solved It

The Bottom Line

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