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Fermat two square: An odd prime p is expressible as

${\displaystyle p=x^{2}+y^{2},\,}$

with $x, y$ integers, if and only if

${\displaystyle p\equiv 1{\pmod {4}}.}$

Lagrange four square: Every positive integer can be written as the sum of at most four squares.

I know there are many proofs for these theorems, but my question is that how it is possible to prove these theorems by Hardy-Littlewood circle method. If not possible to prove it completely (for all numbers stated in theorems) how can we prove them for large enough numbers are possible? Is there any source that is already proved there?

Thanks.

Fermat two square: An odd prime p is expressible as

${\displaystyle p=x^{2}+y^{2},\,}$

with $x, y$ integers, if and only if

${\displaystyle p\equiv 1{\pmod {4}}.}$

Lagrange four square: Every positive integer can be written as the sum of at most four squares.

Legendre's three-square: a natural number can be represented as the sum of three squares of integers

${\displaystyle n=x^{2}+y^{2}+z^{2}}$

if and only if $n$ is not of the form ${\displaystyle n=4^{a}(8b+7)}$ for integers a and b.

I know there are many proofs for these theorems, but my question is that how it is possible to prove these theorems by Hardy-Littlewood circle method. If not possible to prove it completely (for all numbers stated in theorems) how can we prove them for large enough numbers are possible? Is there any source that is already proved there?

Thanks.