In section 11 of this [paper](http://matwbn.icm.edu.pl/ksiazki/aa/aa48/aa4821.pdf) 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. 



Elaboration of a certain section of a paper by Thanigasalam
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.


