2026/05/04 by Ethan Simpson Lee, Ethan S. Lee, Rowan O'Clarey
Mathematics · #Analytic Number Theory Research #Dirichlet distribution #Integer (computer science) #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Prime (order theory) #Prime number #Riemann hypothesis #math.NT #msc:11A07 #msc:11P32 #msc:11Y35
paper · pdf · doi:10.1007/s00013-026-02275-6
9 pages, feedback warmly welcomed
arxiv created 2026/05/04 · openalex publication_date 2026/08/05 · arxiv updated 2026/08/06 · openalex created_date 2026/08/06 · openalex updated_date 2026/08/06
Abstract 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 gt; 24 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>24</mml:mn> </mml:mrow> </mml:math> and prove two results to support this claim. First, we prove the result holds unconditionally for every odd n gt; 24 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>24</mml:mn> </mml:mrow> </mml:math> . Second, assuming the Generalised Riemann Hypothesis for Dirichlet L -functions, we prove the result holds for every n gt; 24 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>24</mml:mn> </mml:mrow> </mml:math> . We also discuss the obstruction which prohibits us from proving the result unconditionally for every n gt; 24 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>24</mml:mn> </mml:mrow> </mml:math> .