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Every sufficiently large natural number can be written as $p+n_1^2+n_2^2$ for some $n_1,n_2$ - natural numbers and $p$ - prime number.

I saw this used in russian article, but it was given as a fact without comment. Does this theorem have a name?

What methods are used in proofs of theorems stating that sufficiently large integers can be written in some forms? The only somehow similar thing I know is the Lagrange's four-square theorem. But if we involve primes in the representation, similar methods can't work I guess

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This was first proved by Hooley, assuming GRH, and later by You. V. Linnik in $1960$ without assuming the Riemann hypothesis. The proof uses non-trivial methods from analytic number theory, and the result is much more difficult than Lagrange's Theorem. There are several results "approaching the Goldbach conjecture", see for example this article.

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