A density theorem for higher-order sums of prime numbers

Michael T. Lacey; Hamed Mousavi; Yaghoub Rahimi; Manasa N. Vempati

doi:10.1007/s40687-025-00597-5

A density theorem for higher-order sums of prime numbers

Michael T. Lacey^*, Hamed Mousavi, Yaghoub Rahimi, Manasa N. Vempati

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

Abstract

Let P be a subset of the primes of lower density strictly larger than 12. 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, and for 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.

Original language	English
Article number	17
Number of pages	16
Journal	Research in the Mathematical Sciences
Volume	13
Issue number	1
DOIs	https://doi.org/10.1007/s40687-025-00597-5
Publication status	Published - 21 Jan 2026

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© The Author(s) 2026.

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AU - Mousavi, Hamed

AU - Rahimi, Yaghoub

AU - Vempati, Manasa N.

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N2 - Let P be a subset of the primes of lower density strictly larger than 12. 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, and for 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.

AB - Let P be a subset of the primes of lower density strictly larger than 12. 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, and for 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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A density theorem for higher-order sums of prime numbers

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