Elaboration of a certain section of a paper by Thanigasalam

Question

In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le 105\Rightarrow H(10)\le 107$? (In this case, $G(k)$ is the minimal $s$ such that all sufficiently large integers are the sums of $s$ $k$th powers of nonnegative integers, and $H(k)$ is the minimal $s$ such that all sufficiently large integers satisfying some local conditions are the sums of $s$ $k$th powers of primes.

Answer 1

The place to look is Section 17 of Thanigasalam's earlier paper, http://matwbn.icm.edu.pl/ksiazki/aa/aa46/aa4611.pdf. In particular, it is not the case that $G(10)\leq 105$ implies directly $H(10)\leq 107$, but rather that the same argument can be used for both.

That is, if one can control the minor arcs using $s_2$ summands additively Hua's lemma style, and $s_1$ summands via Weyl's inequality or similar, then one can deduce via the circle method that $G(k)\leq s_1+2s_2$.

A straightforward adaptation, sketched in the aforementioned Section 17, allows one to also apply this argument to primes, obtaining $H(k)\leq s_1'+2s_2+1$, where $s_1'$ is $s_1$ if $s_1$ is even and $s_1+1$ otherwise.

That $H(10)\leq 107$ is the $s_1=3$ and $s_2=51$ case of this argument.