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February 27, 20260 citations

A system of two Diophantine inequalities with primes

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YDYanjun DongQWQian Wang

Key Points

  • The main goal is to establish conditions under which a system of Diophantine inequalities has prime solutions.
  • Fixed k ≥ 7, define v based on k and constants.
  • Set parameters 1 < d < c < v and conditions on μ and ν.
  • Utilize inequalities involving prime sums to establish bounds.
  • Confirmed conditions yield prime solutions p1,…,pk.
  • Demonstrated specific bounds on real numbers related to primes.
  • Established logarithmic relationships in inequalities.

Abstract

Let k ≥ 7 be fixed, and v = (55k + 556)/(26k + 672),1 0 are real numbers, and that N1, N2 are real numbers satisfying N1 > N′1 and N2 > N′2. Then we prove the system of two Diophantine inequalities |p1c + … + pkc − N1 | < N1−(1/c)(v−c) log195 N1, |p1d + … + pkd − N2 | < N2−(1/d)(v−d)log195 N2 has prime solutions p1,…,pk.

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Cite This Study

Dong et al. (2026) studied this question.

synapsesocial.com/papers/69a13591ed1d949a99abf9bahttps://doi.org/10.1051/ita/2026003/pdf
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