A ternary additive problem (Q387557)

\textit{J. Brüdern} [Math. Scand. 68, No. 1, 27--45 (1991; Zbl 0759.11031)] proved that the number of integers \(n \leq N\) that cannot be written in the form \(n=x^2+y^3+z^5\), (\(x,y,z\) positive integers) is \(O(N^{1-1/30+\varepsilon})\), \(\varepsilon>0\). In this paper the authors study representations of \(n=m^2+p^3+q^5\), where \(m\) is square-free, and \(p\) and \(q\) are primes. They prove that the number of possible exceptions \(n \leq N\), not of this form, is \(O(N^{1-1/45+\varepsilon})\). (They remark that the 45 can be replaced by 30, if there is no condition on \(m\)). For \(n=m^2+p^3+q^k\) they prove the corresponding bounds, for \(k=3\): \(O(N^{1-1/12+\varepsilon})\), and \(k=4\): \(O(N^{1-1/18+\varepsilon})\). They also study the problem with \(m\) not containing any prime divisor smaller than \(c \log n\), for some positive constant \(c\). For the problem \(n=p_1^2+p_2^3+p_3^5+p_4^k\), all \(p_i\) prime, the authors obtain the following new upper bounds for the exceptional sets: \(k=4\): \(O(N^{1-167/5040+\varepsilon})\), and \(k\geq 5\): \(O(N^{1-47/(420\cdot 2^s)+\varepsilon})\), where \(s=\lfloor \frac{k+1}{2}\rfloor\).