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<p>We prove that every sufficiently large integer <i>n</i> can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n &gt; 24\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>24</mn> </mrow> </math></EquationSource> </InlineEquation> and prove two results to support this claim. First, we prove the result holds unconditionally for every odd <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n &gt; 24\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>24</mn> </mrow> </math></EquationSource> </InlineEquation>. Second, assuming the Generalised Riemann Hypothesis for Dirichlet <i>L</i>-functions, we prove the result holds for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n &gt; 24\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>24</mn> </mrow> </math></EquationSource> </InlineEquation>. We also discuss the obstruction which prohibits us from proving the result unconditionally for every <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n &gt; 24\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>24</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On the sum of a prime and a number that is not square-free

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年8月5日

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  • Ethan Simpson Lee,
  • Rowan O’Clarey

摘要

We prove that every sufficiently large integer n can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every \(n > 24\) n > 24 and prove two results to support this claim. First, we prove the result holds unconditionally for every odd \(n > 24\) n > 24 . Second, assuming the Generalised Riemann Hypothesis for Dirichlet L-functions, we prove the result holds for every \(n > 24\) n > 24 . We also discuss the obstruction which prohibits us from proving the result unconditionally for every \(n > 24\) n > 24 .