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

Step 4 of 7: Combining with Lagrange's four-square theorem: every

y^k

falls into place

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In plain words

Take $r=4$ in Hilbert's identity. Lagrange's four-square theorem writes each non-negative integer $y$ as $y=x_1^2+x_2^2+x_3^2+x_4^2$ , so substitution gives $y^k=\sum_i a_i z_i^{2k}$ with integer $z_i$ . This is a useful rational-combination identity for the values $y^k$ , but it is not the hypothesis of Step 2 for exponent $2k$ , which would require representing each integer $n$ itself. The relation is therefore an ingredient in Hilbert's induction, not a standalone proof of $g(2k)<\infty$ .

y^k = \sum_{i=1}^{M} a_i\, z_i^{2k},\qquad z_i = b_{i,1}x_1+\cdots+b_{i,4}x_4,\ \ y=x_1^2+x_2^2+x_3^2+x_4^2

Detailed analysis

Substituting a four-square representation into Hilbert's identity gives $y^k=\sum_i a_i z_i^{2k}$ for every non-negative $y$ . Notice the logical scope carefully: the left side is a $k$ -th power, not an arbitrary integer. Thus Step 2 cannot be applied with target exponent $2k$ from this equation alone. Hilbert's full proof uses this identity together with further induction and auxiliary representations; the next step records that additional induction rather than claiming that the displayed identity by itself finishes the even cases.