# Sums of powers

**Sums of powers** are expressions in which a positive integer is written as a sum of k-th powers of integers, such as squares, cubes, or fourth powers. Two classical problems dominate the subject: [Waring's problem](https://www.edgepedia.org/warings-problem), which asks how many k-th powers are needed to represent every positive integer, and the power-sum problem, which asks for a closed formula for the sum \( 1^{k} + 2^{k} + \cdots + n^{k} \). Both have been studied since antiquity and both remain active research areas.<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup><sup> • </sup><sup>[2](https://old.maa.org/node/139084)</sup>

| Key fact | Detail |
|---|---|
| Waring's statement (1770) | Every positive integer is a sum of four squares, of nine cubes, and of nineteen fourth powers, stated without proof<sup>[3](https://dlmf.nist.gov/27.13)</sup> |
| Hilbert's theorem (1909) | For every \( k \geq 2 \) there exists an \( s = s(k) \) such that every natural number is a sum of \( s \) k-th powers of non-negative integers<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> |
| Exact small values | \( g(2) = 4 \) (Lagrange, 1770) and \( g(3) = 9 \) (Wieferich, 1909)<sup>[3](https://dlmf.nist.gov/27.13)</sup><sup> • </sup><sup>[4](https://mathworld.wolfram.com/WaringsProblem.html)</sup> |
| Lower bound for \( g(k) \) | \( g(k) \geq 2^{k} + \lfloor 3^{k}/2^{k} \rfloor - 2 \) for all \( k \geq 2 \), with equality for \( 4 \leq k \leq 200{,}000 \)<sup>[3](https://dlmf.nist.gov/27.13)</sup> |
| Asymptotic formula | Hardy and Littlewood (1928) showed \( J_{s,k}(N) = AN^{s/k-1} + O(N^{s/k-1-\gamma}) \) for \( s \geq (k-2)2^{k-1} + 5 \)<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> |
| Faulhaber's formula (1631) | Sums of odd powers are polynomials in \( N = n(n+1)/2 \); for example \( \sum_{k=1}^{n} k^{3} = N^{2} \)<sup>[5](https://mathworld.wolfram.com/FaulhabersFormula.html)</sup> |
| General power-sum formula | \( \sum_{k=1}^{n} k^{p} = \frac{1}{p+1} \sum_{i=1}^{p+1} (-1)^{\delta_{ip}} \binom{p+1}{i} B_{p+1-i} \, n^{i} \), where \( B_i \) is the i-th Bernoulli number<sup>[5](https://mathworld.wolfram.com/FaulhabersFormula.html)</sup> |

## Waring's problem

Waring's problem was formulated in 1770 by Edward Waring, who asserted that every natural number is a sum of 4 squares, of 9 cubes, and of 19 fourth powers. Put generally, the assertion states that for each exponent \( k \geq 2 \) a number \( s = s(k) \) can be found, determined solely by \( k \), with the property that every natural number can be written as \( s \) k-th powers of non-negative integers.<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> Waring stated this without proof and with limited numerical evidence.<sup>[3](https://dlmf.nist.gov/27.13)</sup>

The square case had already been settled: Lagrange proved in 1770 that \( g(2) = 4 \), the four-square theorem. During the following 139 years the existence of \( g(k) \) was established for \( k = 3, 4, 5, 6, 7, 8, 10 \). [David Hilbert](https://www.edgepedia.org/david-hilbert) gave the first general solution in 1909, proving that \( g(k) \) exists for every \( k \), though without determining its numerical value; for this reason the problem is sometimes called the Hilbert–Waring problem.<sup>[3](https://dlmf.nist.gov/27.13)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> For cubes, Wieferich proved \( g(3) = 9 \) in 1909, and showed that some integers require all nine cubic summands.<sup>[4](https://mathworld.wolfram.com/WaringsProblem.html)</sup><sup> • </sup><sup>[6](https://www.spektrum.de/lexikon/mathematik/waringsches-problem/10854)</sup> For fourth powers, Liouville proved in 1859, using [Lagrange's four-square theorem](https://www.edgepedia.org/lagranges-four-square-theorem) and a polynomial identity, that \( g(4) \leq 53 \).<sup>[4](https://mathworld.wolfram.com/WaringsProblem.html)</sup>

**The exact value of \( g(k) \).** A lower bound comes from a simple construction: \( g(k) \geq 2^{k} + \lfloor 3^{k}/2^{k} \rfloor - 2 \) for all \( k \geq 2 \). Equality holds for \( 4 \leq k \leq 200{,}000 \), and if \( 3^{k} = q2^{k} + r \) with \( 0 < r < 2^{k} \), equality holds whenever \( r + q \leq 2^{k} \), a condition satisfied with at most a finite number of exceptions. The exact value of \( g(k) \) is now known for every \( k \leq 200{,}000 \).<sup>[3](https://dlmf.nist.gov/27.13)</sup> A conjectural formula \( g(k) = 2^{k} + \lfloor (3/2)^{k} \rfloor - 2 \) appears in the survey literature.<sup>[7](https://www.ams.org/journals/bull/2001-38-01/S0273-0979-00-00890-9/S0273-0979-00-00890-9.pdf)</sup>

## The function G(k) and large numbers

A related but different quantity, \( G(k) \), is the smallest \( m \) such that every sufficiently large integer, rather than every integer, is a sum of \( m \) k-th powers. Its existence follows from that of \( g(k) \) because \( G(k) \leq g(k) \), but \( G(k) \) is generally much smaller for large \( k \). Only the values \( G(2) = 4 \) and \( G(4) = 16 \) are known exactly; for other exponents upper bounds are known, for example \( G(3) \leq 7 \), \( G(5) \leq 23 \), \( G(6) \leq 36 \), \( G(7) \leq 53 \), and \( G(8) \leq 73 \).<sup>[3](https://dlmf.nist.gov/27.13)</sup> The value \( G(4) = 16 \) is due to H. Davenport (1939) and \( G(3) = 7 \) to Yu. V. Linnik (1942).<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup>

**Upper bounds for \( G(k) \).** It is known that \( G(k) \geq k + 1 \). Vinogradov proved in 1934, using his method of trigonometric sums, that \( G(k) \leq 3k(\ln k + 9) \).<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> [Heilbronn](https://www.edgepedia.org/heilbronn) (1936) improved results of Vinogradov to \( G(n) \leq 6n \ln n + [4 + 3\ln(3 + 2/n)]n + 3 \); Karatsuba (1985) showed \( G(k) < 2k \ln k + 2k \ln \ln k + 6k \) for \( k > 224791 \); and Wooley (1992) proved \( G(k) < (1 + c)k \ln k \) for large \( k \) and any positive \( c \).<sup>[4](https://mathworld.wolfram.com/WaringsProblem.html)</sup> The best bound commonly cited is Wooley's \( G(k) \leq k \log k + k \log \log k + O(k) \).<sup>[8](https://mathoverflow.net/questions/345079/what-is-the-best-known-upper-bound-for-gk-in-warings-problem)</sup>

Hardy and Littlewood conjectured in 1925 that \( G(k) < 2k + 1 \) when \( k \) is not a power of 2, and that \( G(k) \leq 4k \) when \( k \) is a power of 2; the most that is known, as of 2009, is \( G(k) < c \, k \ln k \) for some constant \( c \).<sup>[3](https://dlmf.nist.gov/27.13)</sup> For cubes specifically, Deshouillers and colleagues conjectured in 2000 that 7373170279850 is the largest integer that cannot be expressed as a sum of four non-negative cubes.<sup>[4](https://mathworld.wolfram.com/WaringsProblem.html)</sup>

## The circle method and the asymptotic formula

The main tool for quantitative results is the [Hardy–Littlewood circle method](https://www.edgepedia.org/hardy-littlewood-circle-method). Hardy and Littlewood wrote an integral expression for the number of ways to write \( N \) as a sum of \( s \) k-th powers, estimated it, and showed that for large enough \( s \) and \( N \) the count is positive; since it is a whole number, it must be at least 1, giving a representation.<sup>[9](https://people.maths.ox.ac.uk/outreach/neale/Waring_short.html)</sup> Concretely, they demonstrated in 1928 that for \( s \geq (k-2)2^{k-1} + 5 \) the number \( J_{s,k}(N) \) of solutions of \( x_1^k + \cdots + x_s^k = N \) in non-negative integers satisfies an asymptotic formula

\[ J_{s,k}(N) = A N^{s/k - 1} + O(N^{s/k - 1 - \gamma}), \]

where \( A = A(N) \geq c_0 > 0 \) and \( c_0, \gamma > 0 \) are constants. Consequently, if \( N \geq N_0(k) \), the equation has a solution.<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> An elementary proof of Waring's problem, avoiding the analytic machinery, was given in 1942 by Linnik.<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup>

**Modern refinements.** Research on Waring's problem has focused on three quantities: the least \( s \) for which solutions exist for all sufficiently large \( N \), denoted \( G(k) \); the least \( s \) for all \( N \), denoted \( g(k) \); and the least \( s \) for which the asymptotic formula holds, denoted \( \widetilde{G}(k) \), with \( \widetilde{G}(k) \geq G(k) \) and \( g(k) \geq G(k) \).<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> Bounds for \( \widetilde{G}(k) \) have fallen sharply through work on [Vinogradov's mean value theorem](https://www.edgepedia.org/vinogradovs-mean-value-theorem): Hardy and Littlewood's Weyl-differencing bound \( \widetilde{G}(k) \leq (k-2)2^{k-1} + 5 \) was refined by Hua (1938) to \( \widetilde{G}(k) \leq 2^{k} + 1 \), and by Vaughan (1986) to \( \widetilde{G}(k) \leq 2^{k} \) for \( k \geq 3 \).<sup>[10](https://arxiv.org/abs/1404.3508)</sup> Trevor D. Wooley's efficient congruencing method, which exploits the translation invariance of associated Diophantine systems, established the conjectured asymptotic formula for sums of \( s \) k-th powers whenever \( s \geq 2k^{2} + 2k - 3 \).<sup>[11](https://arxiv.org/abs/1101.0574)</sup><sup> • </sup><sup>[10](https://arxiv.org/abs/1404.3508)</sup>

## Faulhaber's formula and power sums

The second strand of the subject concerns finite sums \( S_p(n) = \sum_{k=1}^{n} k^{p} \). The quest for closed formulas, motivated largely by the aim of finding areas and volumes, runs from the Pythagoreans and [Archimedes](https://www.edgepedia.org/archimedes) through Nicomachus, Āryabhata, al-Karajī, ibn al-Haytham, Fermat, Pascal, Bernoulli, and Euler.<sup>[2](https://old.maa.org/node/139084)</sup> Blaise Pascal produced the first explicit recursive formula for sums of powers in any arithmetic progression using binomial coefficients, provable by mathematical induction.<sup>[12](https://old.maa.org/sites/default/files/images/upload_library/46/Pengelley_projects/Project-2/sums-of-powers-project.pdf)</sup>

In his *Academia Algebrae* of 1631, Johann Faulhaber published formulas for power sums up to the 17th power, derived without indicating his methods. His results for odd powers express the sums as polynomials in \( N = n(n+1)/2 \), the triangular number: \( \sum k = N \), \( \sum k^{3} = N^{2} \) (a case known as Nicomachus's theorem), \( \sum k^{5} = \frac{1}{3}(4N^{3} - N^{2}) \), and so on up to \( \sum k^{17} \).<sup>[5](https://mathworld.wolfram.com/FaulhabersFormula.html)</sup> [Faulhaber](https://www.edgepedia.org/faulhaber) believed analogous polynomials would continue for all odd powers; a rigorous proof was first published by Jacobi in 1834.<sup>[5](https://mathworld.wolfram.com/FaulhabersFormula.html)</sup>

**Bernoulli numbers.** Jakob Bernoulli collected the formulas for powers 1 through 10 in his *Ars conjectandi* (1713), noticed the pattern in the coefficients, and arrived at the sequence now called the [Bernoulli numbers](https://www.edgepedia.org/bernoulli-numbers). The general closed form, Faulhaber's formula, is

\[ \sum_{k=1}^{n} k^{p} = \frac{1}{p+1} \sum_{i=1}^{p+1} (-1)^{\delta_{ip}} \binom{p+1}{i} B_{p+1-i} \, n^{i}, \]

with \( \delta_{ip} \) the [Kronecker delta](https://www.edgepedia.org/kronecker-delta) and \( B_i \) the i-th [Bernoulli number](https://www.edgepedia.org/bernoulli-number).<sup>[5](https://mathworld.wolfram.com/FaulhabersFormula.html)</sup><sup> • </sup><sup>[13](https://mathworld.wolfram.com/PowerSum.html)</sup> Faulhaber himself was not aware of Bernoulli numbers.<sup>[5](https://mathworld.wolfram.com/FaulhabersFormula.html)</sup> The formula is efficient: Bernoulli reported that with the help of his table it took him less than a quarter of an hour to find that the tenth powers of the first 1000 numbers sum to 91,409,924,241,424,243,424,241,924,242,500.<sup>[14](https://old.maa.org/press/periodicals/convergence/sums-of-powers-of-positive-integers-conclusion)</sup> The same sums admit an analytic expression \( S_p(n) = \zeta(-p) - \zeta(-p, 1+n) \) in terms of the Riemann zeta and Hurwitz zeta functions.<sup>[13](https://mathworld.wolfram.com/PowerSum.html)</sup>

## Variants and related problems

Many generalizations of Waring's problem exist: the variables may run through a subset of the natural numbers, \( N \) may be represented by polynomials \( f_1(x_1), \ldots, f_s(x_s) \) rather than monomials, or the equation may be replaced by a congruence.<sup>[1](https://encyclopediaofmath.org/wiki/Waring_problem)</sup> One active variant, the [Waring–Goldbach problem](https://www.edgepedia.org/waring-goldbach-problem), asks for representations of integers as sums of k-th powers of primes; Brüdern, Kawada, and Wooley proved unconditionally that infinitely many P-2-numbers are sums of \( 2k + 1 \) positive integral k-th powers, and that on the Generalized Riemann Hypothesis infinitely many primes are sums of \( 2\lfloor 4k/3 \rfloor \) such powers.<sup>[15](https://research-information.bris.ac.uk/en/publications/additive-representation-in-thin-sequences-vi-representing-primes-/)</sup> Another line studies sums of two k-th powers: Skinner and Wooley quantified \( \nu_k(x) \), the number of integers up to \( x \) representable as a sum of two non-negative k-th powers in more than one way, in connection with the conjecture that for \( k \geq 5 \) no integer has two essentially different such representations.<sup>[16](https://www.math.purdue.edu/~twooley/publ/1995%20tkp.pdf)</sup> Mixed exponents are covered by [Freiman's theorem](https://www.edgepedia.org/freimans-theorem), made effective by Brüdern and Wooley: when \( \sum k_i^{-1} \) diverges, all large integers are represented as \( x_1^{k_j} + x_2^{k_{j+1}} + \cdots \).<sup>[17](https://arxiv.org/abs/2302.12920)</sup>

## References

1. [Waring problem, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Waring_problem)
2. [Figurate Numbers and Sums of Numerical Powers: Fermat, Pascal, Bernoulli, MAA Convergence](https://old.maa.org/node/139084)
3. [DLMF §27.13: Additive Number Theory, NIST](https://dlmf.nist.gov/27.13)
4. [Waring's Problem, Wolfram MathWorld](https://mathworld.wolfram.com/WaringsProblem.html)
5. [Faulhaber's Formula, Wolfram MathWorld](https://mathworld.wolfram.com/FaulhabersFormula.html)
6. [Waringsches Problem, Lexikon der Mathematik, Spektrum](https://www.spektrum.de/lexikon/mathematik/waringsches-problem/10854)
7. [Review of Vaughan's The Hardy–Littlewood Method, AMS Bulletin 38 (2001)](https://www.ams.org/journals/bull/2001-38-01/S0273-0979-00-00890-9/S0273-0979-00-00890-9.pdf)
8. [Best known upper bound for G(k), MathOverflow](https://mathoverflow.net/questions/345079/what-is-the-best-known-upper-bound-for-gk-in-warings-problem)
9. [Vicky Neale, Waring's problem for beginners, University of Oxford](https://people.maths.ox.ac.uk/outreach/neale/Waring_short.html)
10. [T. D. Wooley, Translation invariance, exponential sums, and Waring's problem, arXiv:1404.3508](https://arxiv.org/abs/1404.3508)
11. [T. D. Wooley, Vinogradov's mean value theorem via efficient congruencing, arXiv:1101.0574](https://arxiv.org/abs/1101.0574)
12. [Sums of Numerical Powers in Discrete Mathematics, MAA/Pengelley project](https://old.maa.org/sites/default/files/images/upload_library/46/Pengelley_projects/Project-2/sums-of-powers-project.pdf)
13. [Power Sum, Wolfram MathWorld](https://mathworld.wolfram.com/PowerSum.html)
14. [Janet Beery, Sums of Powers of Positive Integers – Conclusion, MAA Convergence](https://old.maa.org/press/periodicals/convergence/sums-of-powers-of-positive-integers-conclusion)
15. [J. Brüdern, K. Kawada, T. D. Wooley, Additive representation in thin sequences, VI, Glasgow Mathematical Journal 44 (2002)](https://research-information.bris.ac.uk/en/publications/additive-representation-in-thin-sequences-vi-representing-primes-/)
16. [C. M. Skinner, T. D. Wooley, Sums of two kth powers](https://www.math.purdue.edu/~twooley/publ/1995%20tkp.pdf)
17. [J. Brüdern, T. D. Wooley, On Waring's problem: beyond Freiman's theorem, arXiv:2302.12920](https://arxiv.org/abs/2302.12920)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Diophantine problems and approximation › Linear and additive Diophantine equations*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license