A quaternary Diophantine inequality with prime numbers of a special form (Q6185046)

Motivated by the Waring-Goldbach problem, which concerns the solvability of the following Diophantine equation in prime variables \(p_1,\dots, p_s\), \[ p_1^k+\cdots+p_s^k=N, \] the authors of the present paper prove that for \(N\) sufficiently large real number, for \(1<c<4803/4040\) and for any arbitrary large number \(E>0\), the Diophantine inequality \[ \left|p_1^c+p_2^c+p_3^c+p_4^c-N\right|<(\log N)^{-E} \] is solvable in prime variables \(p_1,p_2,p_3,p_4\) satisfying for \(i=1,2,3,4\), \[ \Omega(p_i+2)\leq \left\lfloor\frac{31540280}{12007500-10100000c}\right\rfloor, \] where \(\Omega(n)\) denotes the total number of prime factors of \(n\) counting with multiplicity. Moreover, letting \(c\to 1\), they imply that all the numbers \(p_i+2\) has at most \(16\) prime factors.