2026/02/28 by Marc Chamberland, Armin Straub
Arts and Humanities · Mathematics · Psychology · #Art, Technology, and Culture #Computation #Conjecture #Goldbach's conjecture #History and Theory of Mathematics #Jungian Analytical Psychology #Number theory #Riemann hypothesis #Twin prime #Upper and lower bounds
paper · doi:10.1080/00029890.2026.2694309
published in American Mathematical Monthly, 1-11 (Taylor & Francis)
openalex publication_date 2026/07/24 · openalex created_date 2026/07/25 · openalex updated_date 2026/07/26
The world of primes has many gaps between evidence and theorems. Here, we review Legendre’s conjecture on primes between consecutive squares and recent progress on the weaker question of primes between consecutive larger powers. Assuming the Riemann hypothesis (RH), we observe that a recent result of Emanuel Carneiro, Micah Milinovich and Kannan Soundararajan, combined with a large-scale computation by Jonathan Sorenson and Jonathan Webster, implies the existence of primes between x2+δ and (x+1)2+δ for all real x≥1 when δ≥1/4. For smaller values of δ>0, we provide an explicit bound x0=x0(δ) such that primes exist in these intervals whenever x≥x0 (again assuming RH). We conclude with an application to Mills-type prime-generating constants.