# Waring–Goldbach problem

The Waring–Goldbach problem asks whether every sufficiently large natural number satisfying certain congruence conditions can be written as a sum of a bounded number of kth powers of prime numbers, and how few summands are needed. It merges two classical questions of additive number theory: [Waring's problem](https://www.edgepedia.org/warings-problem), which concerns sums of kth powers of integers, and Goldbach's problem, which concerns sums of primes. The central quantity is H(k), the least number of prime kth powers required.

| Key fact | Detail |
|---|---|
| Subject | Representation of large integers as sums of kth powers of primes, for each fixed exponent k ≥ 1<sup>[1](https://math.ntnu.edu.tw/~maco/macobook/study/reach1.pdf)</sup> |
| Central function | H(k), the least s such that every sufficiently large n ≡ s (mod K(k)) is a sum of s kth powers of primes<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup> |
| Necessary condition | Any sum of s kth powers of primes exceeding k + 1 satisfies n ≡ s (mod K(k)); for example K(4) = 240 and K(5) = 2<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup> |
| Expected answer | A formal Hardy–Littlewood method argument suggests s ≥ k + 1 should suffice, subject to the local congruence conditions<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup> |
| Known bounds | H(1) ≤ 3, H(2) ≤ 5, H(3) ≤ 9; Kawada and Wooley proved H(4) ≤ 14 and H(5) ≤ 21<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup><sup> • </sup><sup>[3](https://www.cambridge.org/core/journals/proceedings-of-the-london-mathematical-society/article/abs/on-the-waringgoldbach-problem-for-fourth-and-fifth-powers/6C0010F7550204C9D0B42A476FF6EC24)</sup> |
| Large k | Kumchev and Wooley showed H(k) ≤ (4k − 2) log k + k − 7 for large k, the first improvement on Hua's bound from the 1940s<sup>[4](https://arxiv.org/abs/1510.00982)</sup> |

## Origin and definition

The problem grew out of Vinogradov's 1937 proof, unconditional on any unproven hypothesis about zeros of Dirichlet L-functions, that every sufficiently large odd number is a sum of three primes. In the notation of the field this result reads H(1) ≤ 3. Shortly after that proof appeared, Vinogradov himself and Hua Loo-Keng began studying Waring's problem with prime variables, the question now called the Waring–Goldbach problem, and generalized the asymptotic formula to kth powers for all k ≥ 1<sup>[5](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2285-01.pdf)</sup><sup> • </sup><sup>[6](https://www.researchgate.net/publication/2116657_An_invitation_to_additive_prime_number_theory)</sup>. Hua's 1938 paper *Some results in the additive prime-number theory* in the Quarterly Journal of Mathematics was an early landmark of this program<sup>[7](https://academic.oup.com/qjmath/article-abstract/os-9/1/68/1566008)</sup>.

Why congruence conditions appear. For a prime p, let θ = θ(k, p) be the integer with p<sup>θ</sup> dividing k but p<sup>θ+1</sup> not dividing k, and define γ(k, p) = θ + 2 when p = 2 and θ > 0, and θ + 1 otherwise. Then K(k) is the product of p<sup>γ</sup> over primes p with (p − 1) dividing k. Whenever n is a sum of s kth powers of primes greater than k + 1, one necessarily has n ≡ s (mod K(k)); for k = 4 this modulus is 240, and for k = 5 it is 2<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup>. Following Hua, H(k) is defined as the least integer s such that every sufficiently large positive integer congruent to s modulo K(k) is a sum of s kth powers of primes<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup>. Further congruence conditions can arise from primes p with (p − 1) not dividing k<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup>.

## The circle method and the expected answer

The main tool is the [Hardy–Littlewood circle method](https://www.edgepedia.org/hardy-littlewood-circle-method), which splits an integral over a circle into major arcs, where the generating function is large and can be approximated, and minor arcs, where it must be shown to be small. The major-arc contribution produces a product of local densities, the singular series, which measures arithmetic obstructions modulo each prime; if any local density vanishes, the expected count is zero, which is exactly how the method detects the necessary congruence conditions<sup>[8](https://interactiveprimer.com/concepts/mathematics/number-theory/analytic-number-theory/circle-method/the-circle-method)</sup>. Progress on the problem has largely consisted of sharper estimates for exponential sums over primes on the minor arcs<sup>[3](https://www.cambridge.org/core/journals/proceedings-of-the-london-mathematical-society/article/abs/on-the-waringgoldbach-problem-for-fourth-and-fifth-powers/6C0010F7550204C9D0B42A476FF6EC24)</sup>.

A formal application of the method suggests that whenever s ≥ k + 1 and n is a large natural number satisfying the necessary local conditions, n should be representable as a sum of s kth powers of primes<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup>. For k = 4 and k = 5 this heuristic points to H(4) = 5 and H(5) = 10, based on congruence conditions modulo 13 and 11 respectively, values far below what has been proved<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup>.

## Bounds for H(k)

By the middle of the twentieth century, work of Vinogradov, Hua, and Davenport had established H(1) ≤ 3, H(2) ≤ 5, H(3) ≤ 9, H(4) ≤ 15, H(5) ≤ 25, H(6) ≤ 37, H(7) ≤ 55, and H(8) ≤ 75. For k ≤ 4 these bounds remained the sharpest available as of 2001. For k > 5, techniques developed in the mid-1980s by Thanigasalam and Vaughan gave superior bounds, including H(5) ≤ 23, H(6) ≤ 33, H(7) ≤ 47, and H(8) ≤ 63<sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup>.

**Kawada and Wooley, 2001.** Koichi Kawada of Iwate University and Trevor D. Wooley proved that every sufficiently large integer congruent to 14 modulo 240 is a sum of 14 fourth powers of primes, so H(4) ≤ 14, improving the previous bound of 15 that followed from work of Davenport. They also proved H(5) ≤ 21, so that every sufficiently large odd integer is a sum of 21 fifth powers of primes, improving the bound of 23 due to Thanigasalam<sup>[3](https://www.cambridge.org/core/journals/proceedings-of-the-london-mathematical-society/article/abs/on-the-waringgoldbach-problem-for-fourth-and-fifth-powers/6C0010F7550204C9D0B42A476FF6EC24)</sup><sup> • </sup><sup>[2](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)</sup>. Their proofs combined estimates from exponential sums over primes with the linear sieve, rather than the conventional route that sacrifices one or two variables to a mean value estimate over unrestricted integers; for the fifth powers they applied a switching principle to a related problem with almost primes to reach a conclusion involving primes alone<sup>[3](https://www.cambridge.org/core/journals/proceedings-of-the-london-mathematical-society/article/abs/on-the-waringgoldbach-problem-for-fourth-and-fifth-powers/6C0010F7550204C9D0B42A476FF6EC24)</sup>.

**Large exponents.** Hua's classical bound from the 1940s, H(k) ≤ 4k log k + 2k log log k + O(k), stood for over half a century<sup>[5](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2285-01.pdf)</sup>. In 2015, Angel V. Kumchev and Trevor D. Wooley applied new estimates for Weyl-type exponential sums arising from progress on [Vinogradov's mean value theorem](https://www.edgepedia.org/vinogradovs-mean-value-theorem) to prove that H(k) ≤ (4k − 2) log k + k − 7 when k is large, the first improvement on Hua's bound; their computations suggest the bound holds once k exceeds 64, and they also gave explicit bounds for 8 ≤ k ≤ 20<sup>[4](https://arxiv.org/abs/1510.00982)</sup>.

## Variants

**Almost equal summands.** One variant requires the prime summands to be close in size, asking for representations n = p₁ᵏ + ⋯ + pₛᵏ with all primes in a short interval around x = (n/s)<sup>1/k</sup>. Using a transference principle, Salmensuu showed in 2020 that whenever s is suitably large depending on k ≥ 2, every sufficiently large n satisfying some congruence conditions admits such a representation with primes in [x − x<sup>θ</sup>, x + x<sup>θ</sup>] and θ = 0.525 + ε, improving all previously known values of θ<sup>[9](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12019)</sup>. Related work has combined estimates for exponential sums over primes in short intervals with the circle method for this purpose<sup>[10](https://arxiv.org/abs/1412.2743)</sup>.

**Mixed powers.** The problem extends to representations mixing different exponents. Brüdern showed in 1987 that for every large integer n, the equation n = x₁² + x₂² + x₃³ + x₄⁴ + x₅⁵ + x₆⁶ + x₇⁶ has solutions in non-negative integers. The Waring–Goldbach analog, with all variables prime, was established in recent work: every sufficiently large odd integer is a sum of two squares, a cube, a fourth power, a fifth power, and two sixth powers of primes, and every sufficiently large odd integer not divisible by 3 is a sum of two squares, three fourth powers, a fifth power, and a sixth power of primes<sup>[11](https://link.springer.com/article/10.1007/s11139-026-01340-6)</sup>. Other results of this type include representations as one square, four cubes, and higher powers of primes, with one variable allowed to be an almost-prime, a number with a bounded number of prime factors<sup>[12](https://arxiv.org/abs/1708.04484)</sup>.

**Density versions.** Work in the style of Roth's theorem has produced density analogs: any subset of the dth powers of primes with positive relative density contains nontrivial solutions to translation-invariant linear equations in d² + 1 or more variables, with explicit quantitative bounds<sup>[13](https://arxiv.org/abs/1602.04012)</sup>.

## Relation to the parent problems

The Waring–Goldbach problem sits between Waring's problem and [Goldbach's conjecture](https://www.edgepedia.org/goldbachs-conjecture). Waring's problem, stated by Edward Waring in 1770, asks for the least number g(k) of integer kth powers needed to represent every positive integer; Hilbert proved in 1909 that g(k) exists for every k<sup>[14](https://dlmf.nist.gov/27.13)</sup>. Goldbach's conjecture, from 1742, asserts that every even integer greater than 4 is the sum of two odd primes and remains undecided<sup>[14](https://dlmf.nist.gov/27.13)</sup>. The Waring–Goldbach problem replaces the integer variables of Waring's problem with primes, so the k = 1 case is the ternary Goldbach problem, and the methods developed for one problem have repeatedly fed the other: Vinogradov's method of estimating exponential sums over primes, created for the three-primes theorem, shaped research on additive problems involving primes generally<sup>[5](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2285-01.pdf)</sup>.

## References

1. [The Waring-Goldbach problem asks for the possibility of expressing natural numbers as sums of d-th powers of primes (study notes, NTNU)](https://math.ntnu.edu.tw/~maco/macobook/study/reach1.pdf)
2. [K. Kawada and T. D. Wooley, On the Waring-Goldbach problem for fourth and fifth powers (PDF)](https://www.math.purdue.edu/~twooley/publ/2001%20wgp45.pdf)
3. [K. Kawada and T. D. Wooley, On the Waring–Goldbach problem for fourth and fifth powers, Proc. London Math. Soc. 83 (2001)](https://www.cambridge.org/core/journals/proceedings-of-the-london-mathematical-society/article/abs/on-the-waringgoldbach-problem-for-fourth-and-fifth-powers/6C0010F7550204C9D0B42A476FF6EC24)
4. [A. V. Kumchev and T. D. Wooley, On the Waring–Goldbach problem for eighth and higher powers, arXiv:1510.00982](https://arxiv.org/abs/1510.00982)
5. [K. Kawada, On the Waring–Goldbach problem: A survey, RIMS Kôkyûroku 2285](https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2285-01.pdf)
6. [An invitation to additive prime number theory (ResearchGate)](https://www.researchgate.net/publication/2116657_An_invitation_to_additive_prime_number_theory)
7. [L. K. Hua, Some results in the additive prime-number theory, Quart. J. Math. os-9 (1938), 68–80](https://academic.oup.com/qjmath/article-abstract/os-9/1/68/1566008)
8. [The circle method (Interactive Primer)](https://interactiveprimer.com/concepts/mathematics/number-theory/analytic-number-theory/circle-method/the-circle-method)
9. [J. Salmensuu, On the Waring–Goldbach problem with almost equal summands, Mathematika 66 (2020)](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12019)
10. [B. Huang, Exponential sums over primes in short intervals and an application to the Waring–Goldbach problem, arXiv:1412.2743](https://arxiv.org/abs/1412.2743)
11. [On some Waring–Goldbach problems, The Ramanujan Journal](https://link.springer.com/article/10.1007/s11139-026-01340-6)
12. [Waring-Goldbach Problem: One Square, Four Cubes and Higher Powers, arXiv:1708.04484](https://arxiv.org/abs/1708.04484)
13. [Roth–Waring–Goldbach, arXiv:1602.04012](https://arxiv.org/abs/1602.04012)
14. [DLMF §27.13: Additive Number Theory, NIST](https://dlmf.nist.gov/27.13)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Analytic number theory › Additive number theory › Waring's problem and Hilbert–Waring theory*

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