Abstract
In this paper, we show that for any fixed 1 < c < 967/805, every sufficiently large positive number N, and a small constant 𝜀 > 0, the Diophantine inequality \( \left|{p}_1^c+{p}_2^c+{p}_3^c+{p}_4^c-N\right|<\varepsilon \) has a solution in prime numbers p1, p2, p3, p4,such that p1 = x2 + y2 + 1.
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Dimitrov, S.I. The quaternary Piatetski-Shapiro inequality with one prime of the form p = x2 + y2 + 1. Lith Math J 62, 170–191 (2022). https://doi.org/10.1007/s10986-022-09554-z
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DOI: https://doi.org/10.1007/s10986-022-09554-z