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January 23, 2026Research in the Mathematical Sciences0 citationsOpen Access

A density theorem for higher-order sums of prime numbers

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YOYumeng OuHMHamed MousaviYRYaghoub Rahimi

Key Points

  • To establish that every sufficiently large even integer can be expressed as the sum of four primes from a defined dense subset of primes.
  • Analyzed subsets of primes with lower density greater than 1/2.
  • Utilized elementary combinatorial lemmas to derive results.
  • Developed frameworks for k-summands with k greater than or equal to 4.
  • Demonstrated that every large even integer can be written as a sum of four primes from the subset P.
  • Extended previous findings related to sums of two and three primes to higher-order sums.
  • Provided new insights into combinatorial approaches in number theory.

Abstract

Abstract Let P be a subset of the primes of lower density strictly larger than 12 1 2. Then, every sufficiently large even integer is a sum of four primes from the set P. We establish similar results for k -summands, with k 4 k ⩾ 4, and for k 4 k ⩾ 4 distinct subsets of primes. This extends the work of H. Li, H. Pan, as well as X. Shao on sums of three primes, and A. Alsteri and X. Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.

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Cite This Study

Ou et al. (2026) studied this question.

synapsesocial.com/papers/69730f9fc8125b09b0d1f55ahttps://doi.org/10.1007/s40687-025-00597-5
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