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Demonstrates solutions to a Diophantine inequality involving prime numbers, suggesting new pathways in number theory.
Let 1 < c < (300s + 31)/(200s + 64) be fixed. Assume that N > 0 is a large enough number and ε > 0 is an arbitrarily small constant. This paper establish that the Diophantine inequality |p1c + … + psc − N | < ε has a solution in prime numbers p1,…,ps, where s ≥ 7 and s is a natural number and p1 can be expressed as p1 = m2 + n2 + 1 for some integers m,n.
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Liu et al. (2026) studied this question.
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