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I am looking for problems comparable to the ternary Goldbach problem, which says that every positive odd integer may be written as the sum of three primes. For instance, something of the shape

Is every sufficiently large positive integer the sum of five squares of primes?

I'm looking for problems along these lines, which looks like it might be approachable using the same methodology as Vinogradov/Helfgott's proofs?

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    $\begingroup$ See: mathoverflow.net/questions/491429/…. $\endgroup$ Commented Oct 30 at 20:54
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    $\begingroup$ This question is similar to: Vinogradov's method for sum of more than three primes. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. $\endgroup$ Commented Oct 30 at 20:56
  • $\begingroup$ see Vaughan, The Hardy-Littlewood Method. $\endgroup$ Commented Oct 30 at 22:17
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    $\begingroup$ If $n\equiv2\bmod4$ is a sum of five squares of primes, then $n-12$ is a sum of two squares of primes, and there are infinitely many $n$ for which this is not so. If $n\equiv7\bmod8$, then $n$ is not a sum of five squares of primes. $\endgroup$ Commented Oct 31 at 1:29

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Here is one I stumbled upon by playing around with questions of this sort, only to discover that Zhi-Wei Sun had asked this years earlier, if I remember correctly it was this:

Is every natural number except $216$ a triangular number or the sum of a prime and a triangular number?

But additive number theory is not my field of specialisation, so I do not know whether it can be approached using the methodology you have in mind.

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