Geodesic
On a Diophantine inequality with prime numbers of a special type
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 261-282

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We consider the Diophantine inequality $|p_1^c+p_2^c+p_3^c-N|(\log N)^{-E}$, where $1$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1+2$, $p_2+2$ and $p_3+2$ has at most $[369/(180-168c)]$ prime factors, counted with multiplicity. 
@article{TM_2017_299_a15,
     author = {D. I. Tolev},
     title = {On a {Diophantine} inequality with prime numbers of a special type},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {261--282},
     publisher = {mathdoc},
     volume = {299},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_299_a15/}
}
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D. I. Tolev. On a Diophantine inequality with prime numbers of a special type. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 261-282. http://geodesic.mathdoc.fr/item/TM_2017_299_a15/