An additive equation involving fractional powers (Q2330066)

Let \([x]\) the integer part of the real number \(x\). In this paper the author deals with the equation \([p_1^c]+[p_2^c]=n\), where \(p_1\) and \(p_2\) are primes. He shows that, for any \(c\in (1,14/11)\) and all \(n\in (N/2,N]\) but \(O(N\exp{(-\log^1/6{N})})\) exceptions, this equation is solvable. Let \(\mathcal{R}(n)=\sum_{[p_1^c]+[p_2^c]=n}\log{p_1}\log{p_2}\). He also establishes that, for the same values of \(n\), \(\mathcal{R}(n)=\frac{\Gamma^2(\frac{1}{c}+1)}{\Gamma(\frac{2}{c})} n^{\frac{2}{c}-1} +(N^{\frac{2}{c}-1}\exp{(-\log^{1/6}{N})})\). These results generalize some previously known observations from this area of number theory.