Lagrange's four-square theorem states that every natural number can be represented as the sum of four integer squares.
$$p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}$$ where the four numbers $a_{0},a_{1},a_{2},a_{3}$ are integers (source).
I would like to calculate as few square numbers as possible.
Jacobi's theorem: there are multiple combinations of $[a_{0},a_{1},a_{2},a_{3}]$ for a number (source).
Eduard Wirsing proved that there exists a set of squares S with: $$|S|=O(n^{1/4}\log^{1/4}n)$$ such that every positive integer smaller than or equal n can be written as a sum of at most 4 elements of S (source). Note: I don't have access to this article.
Is it $(\log n)^{1/4}$? Is it $\log_{2}$ or $\ln$ ? Note: it's not $\log_{1/4}(n)$ gives negative numbers. I suppose that the formula is: $${|S|=O(n^{1/4}(\ln n)^{1/4})}$$
Could you confirm me that, by example, for: $n=2^{10}=1024$ $$O(1024^{1/4} \ln(1024)^{1/4})=9,18 => 10$$ Every number between $0$ and $1024$ can be represented by the square numbers between: $0$ and $10$.
$$2^{10} = 1024 = 10^2 + 4^2 + 2^2 + 2^2$$
Is there a better result than $|S|=O(n^{1/4}(\ln n)^{1/4})$?