Every whole number is a sum of a few squares. Try it: 7 = 4 + 1 + 1 + 1 (that's 2^2 + 1^2 + 1^2 + 1^2), 23 = 9 + 9 + 4 + 1, 100 = 64 + 36 = 8^2 + 6^2. No matter which number you name, four squares always suffice — a fact proved by Lagrange in 1770. In that same year Edward Waring, a Cambridge professor, asked the natural sequel: what about cubes? Fourth powers? Fifth? For each exponent k, is there a fixed number of k-th powers that can build every natural number?

Waring conjectured — with no proof, on pure faith — that the answer is always yes: nine cubes suffice for everything, nineteen fourth powers, and so on for each k. It took 139 years before Hilbert proved he was right, and another generation before the the circle method of Hardy and Littlewood turned the qualitative "yes, finitely many" into a precise machine that counts the representations. Waring's problem is the original triumph of that method — the proving ground on which analytic number theory learned to attack additive questions.

The starting point: four squares

Four is also necessary: no smaller number of squares will do for everyone. The obstruction is a class of stubborn numbers. Consider 7. The squares below it are 0, 1, 4; try to hit 7 with just three of them and you are stuck — the best you can manage is 4 + 1 + 1 = 6 or 4 + 4 = 8, never 7. Only when you allow a fourth square does 7 = 4 + 1 + 1 + 1 appear.

In fact every number of the form 4^a(8b + 7) — 7, 15, 23, 28, 31, \dots — genuinely requires all four squares. So the answer for k = 2 is exactly four, and not because we lack cleverness: three squares provably cannot reach these numbers. This tension — a handful of awkward small numbers forcing the count up — is the theme of the whole subject.

Hilbert's theorem: the number always exists

This is a pure existence statement: Hilbert showed g(k) is finite for all k, resolving Waring's conjecture, but his combinatorial proof gave hopeless, astronomically large bounds and no way to compute the true value. Naming the quantity is the first step; pinning it down is the story of the next century.

Two numbers do the pinning, and telling them apart is the single most important idea on this page. Write:

Because G(k) forgives a finite list of exceptions, it is always \le g(k) — and usually much smaller. G(k) is the deeper, more natural quantity; g(k) is essentially a story about a few small numbers.

Why g(k) is dominated by small numbers

Look at cubes. It turns out g(3) = 9: nine cubes suffice for everything. But which numbers actually need nine? Only two of them, in the entire infinite list of naturals:

23 and 239 are the only numbers requiring nine cubes. A slightly larger threshold — numbers needing eight cubes — has just fifteen members, the largest being 454. Beyond 8042, every number is a sum of at most seven cubes, and it is believed (though not fully proven) that beyond a point only four cubes are ever needed. That last figure is G(3) \le 7 at work: the bulk of the numbers are far easier than the two troublemakers 23 and 239 that inflate g(3) to nine.

The same happens for fourth powers, only more dramatically: g(4) = 19, forced by numbers like 79 (which needs nineteen fourth powers, mostly 1s and 16s), yet G(4) = 16 — one of the few G(k) known exactly, proved by Davenport in 1939.

Worked example: three squares fall short, four succeed

Let's nail down why g(2) = 4 and not 3, by pinning the smallest offender. Claim: 7 is not a sum of three squares. The available squares are 0, 1, 4 (since 3^2 = 9 > 7). Enumerate every way to pick three of them:

None equals 7 — it slips between 6 and 8. There is also a clean modular reason: modulo 8 a square is 0, 1, or 4, and no three of those add to 7 \pmod 8. But add a fourth square and the jam clears instantly:

Now a cube example, to feel g(3). Take 23, the champion troublemaker. The cubes available are 1 and 8 (since 3^3 = 27 > 23). To use as few as possible we grab the big cubes first, but two 8s leave 23 - 16 = 7, and 7 must be built from seven 1s:

You cannot do better: a third 8 overshoots (24 > 23), so you are stuck with two 8s and seven 1s. That is exactly why g(3) \ge 9.

Known values of g(k) and G(k)

Here are the small cases side by side. Notice how g(k) races upward while G(k) — where known — lags far behind, and how few of the G(k) are actually settled.

Only G(2) = 4 and G(4) = 16 are known exactly; every other G(k) is trapped between a lower and an upper bound. The upper bounds are exactly what the circle method — and its modern engine, Vinogradov's mean value theorem — keep driving down.

A closed formula for g(k)

The astonishing thing about g(k) is that, being pinned by small awkward numbers, it obeys a clean formula. The worst offender for exponent k is typically the number 2^k\lfloor(3/2)^k\rfloor - 1, which is stuck using only 1s and 2^ks. Working out how many that takes gives:

Check it: k = 2 gives 4 + 2 - 2 = 4 ✓; k = 3 gives 8 + \lfloor 3.375 \rfloor - 2 = 8 + 3 - 2 = 9 ✓; k = 4 gives 16 + \lfloor 5.06 \rfloor - 2 = 16 + 5 - 2 = 19 ✓; k = 5 gives 32 + \lfloor 7.59 \rfloor - 2 = 32 + 7 - 2 = 37 ✓. The formula is known to hold except possibly for finitely many k where (3/2)^k lands suspiciously close to an integer — a condition that is believed never to occur. So g(k), the "all numbers" quantity, is in a sense solved. It is G(k) that remains genuinely open.

The circle method: turning counting into an integral

How does one prove a large number is a sum of s k-th powers? Not by exhibiting the powers — by counting the representations and showing the count is positive. The circle method encodes the count as a single integral. Let e(\theta) = e^{2\pi i \theta} and form the generating (exponential) sum

Raise it to the s-th power: expanding F(\alpha)^s produces a term e\big((m_1^k + \cdots + m_s^k)\,\alpha\big) for every ordered choice of s bases. The number of ways to write N = m_1^k + \cdots + m_s^k is then extracted by the orthogonality of exponentials — integrating against e(-N\alpha) over a full period picks out exactly the terms that sum to N:

The genius is the split. The unit interval is carved into major arcs — tiny neighbourhoods of rationals a/q with small denominator, where F(\alpha) is large and behaves predictably — and the leftover minor arcs, where F(\alpha) is small because the phases e(m^k\alpha) point every which way and cancel.

Major arcs, minor arcs, and the main term

On the major arcs, the exponential sum is close to a product of a local factor at each prime and the reals, and the integral there evaluates to an explicit main term:

Two pieces control it. The singular series \mathfrak{S}(N) is an infinite product over primes measuring whether N is representable modulo every prime power — the local obstructions; when it is bounded below by a positive constant, no congruence blocks a solution. The singular integral (the Gamma-factor piece) is the real-density contribution. Their product, times N^{s/k - 1}, is the expected count — and it grows without bound once s > k.

The whole method succeeds if the minor arcs contribute less than this main term. That is an estimation problem: bound F(\alpha) when \alpha is badly approximable. The classical tool is Weyl's inequality, which gives |F(\alpha)| \ll N^{1/k + \varepsilon}(\cdots) through repeated squaring of the sum; feeding it in shows G(k) \le 2^k + 1 — Hardy and Littlewood's original bound. The sharper the minor-arc bound, the smaller the s you can get away with, and the smaller your bound on G(k).

Vinogradov, Wooley, and the modern collapse of G(k)

The decisive improvement came from bounding not a single sum but its mean value. Vinogradov's mean value theorem controls the average size of

which counts systems of equal power sums. For decades the sharp bound — the Main Conjecture for Vinogradov's mean value — was out of reach. It was finally proved around 2015–2016, by Bourgain, Demeter and Guth (via decoupling in harmonic analysis) and, independently, by Wooley (via his efficient congruencing). The payoff for Waring's problem was immediate and dramatic:

This is the modern face of the century-old circle method: an additive question about integers, settled through the geometry of exponential sums and cutting-edge harmonic analysis.

Seeing the gap

Plotting the two quantities makes the whole moral visible at a glance. g(k) = 2^k + \lfloor(3/2)^k\rfloor - 2 shoots up exponentially — dragged there by a couple of small stubborn numbers for each k. The best known upper bound for G(k), the deep "large-N" quantity, grows only gently. The chasm between the curves is the chasm between "every last number, including the awkward ones" and "every number big enough to behave."

The upper curve is a story about arithmetic accidents; the lower curve is the real mathematics.