Worked solution: Hilbert's existence proof for Waring's problem (1909)

Step 6 of 7: A purely qualitative victory: existence without a bound

In plain words

Stepping back, notice what kind of victory this is. At no point did the proof ever produce a usable numerical value of $M$ (how many terms in Hilbert's identity) or track how these constants grow with $k$ . The argument only shows that finitely many terms suffice — it does not say how many.

This was itself philosophically notable at the time: pure existence proofs, especially ones relying on infinite or continuous constructions (like averaging over a sphere) to conclude something about finite, discrete combinatorics, were part of a broader debate about the nature of mathematical proof that Hilbert himself was a central figure in (a few years earlier, in 1888, his own existence proof of the finite basis theorem in invariant theory had provoked exactly this kind of controversy).

\liminf_{k} \frac{g(k)}{2^k}\ge 1\quad(\text{no explicit bound from Hilbert's method})

Detailed analysis

Hilbert's 1909 proof establishes $g(k)<\infty$ for every $k$ without ever exhibiting a usable formula or numerical bound for $g(k)$ , $M$ , or the constants $a_i,b_{i,j}$ in his identity as functions of $k$ (Wolfram MathWorld and multiple secondary sources describe Hilbert's original argument as involving an especially unwieldy, high-multiplicity multiple integral once fully spelled out; Rademacher and Toeplitz's 1957 exposition The Enjoyment of Mathematics reworked the identity to make it more tractable, still without producing sharp numerical bounds). The first genuinely explicit bound on $g(k)$ came much later, from G. Rieger in 1953, giving an explicit but very weak bound, using a simplification of Hilbert's method rather than Hilbert's original numerics.

This qualitative character reflects a real methodological choice, not a gap: the sphere-averaging identity is fundamentally an existence argument (a positive integral is nonzero, or a positive rational combination exists), and turning it into explicit numbers requires additional, separate combinatorial work that Hilbert did not undertake. The historical context matters here too — existence proofs of this flavor (Hilbert's 1890 finite basis theorem in invariant theory being the most famous earlier example) were controversial among mathematicians who wanted constructive methods, and Hilbert's Waring's-problem proof continued in that same non-constructive spirit.

This qualitative gap is exactly what the 1920 Hardy–Littlewood circle method later filled for large exponents: rather than reproving finiteness, it directly attacks the counting problem analytically (via contour integrals of the generating function $F(z)=\sum_{a\ge0}z^{a^k}$ around the unit circle, split into 'major' and 'minor' arcs), yielding both existence and explicit, often near-optimal, numerical bounds for $g(k)$ and the related quantity $G(k)$ (the number of $k$ -th powers needed for all sufficiently large integers). Vinogradov and later Wooley refined the circle method further, and today $g(k)$ is known exactly for a very wide range of $k$ , conjectured in general by the formula $g(k)=2^k+\lfloor(3/2)^k\rfloor-2$ .

Terms in this step

Non-constructive existence proof: A proof that something exists (here, a finite $g(k)$ ) without exhibiting an explicit value or algorithm to compute it — contrasted with a constructive proof, which would produce the actual number or a method to find it.
Circle method: An analytic technique introduced by Hardy and Littlewood (building on Hardy and Ramanujan's earlier work on partitions), studying a counting problem by integrating a generating function around the unit circle in the complex plane, splitting the circle into arcs near rational points ('major arcs') and everywhere else ('minor arcs').