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<p>Let <i>P</i> be a subset of the primes of lower density strictly larger than <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\frac{1}{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>. Then, every sufficiently large even integer is a sum of four primes from the set <i>P</i>. We establish similar results for <i>k</i>-summands, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\geqslant 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, and for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k \geqslant 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> distinct subsets of primes. This extends the work of H.&#xa0;Li, H.&#xa0;Pan, as well as X.&#xa0;Shao on sums of three primes, and A.&#xa0;Alsteri and X.&#xa0;Shao on sums of two primes. The primary new contributions come from elementary combinatorial lemmas.</p>

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A density theorem for higher-order sums of prime numbers

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月21日

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

摘要

Let P be a subset of the primes of lower density strictly larger than \(\frac{1}{2}\) 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 \(k\geqslant 4\) k ⩾ 4 , and for \(k \geqslant 4\) 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.