Universal Inclusion of Prescribed Primes in 3x3 Magic Squares

2026/04/10 by David Salas, Eloy Timón, Pepa Montero +2 · 1 voice
Mathematics · #math.GM

paper · pdf

arxiv published 2026/04/10 · arxiv updated 2026/06/24

Abstract

We present an integrated version of the global program proving that every prescribed prime \(q₀≥ 5\) occurs in some \(3× 3\) magic square whose nine entries are distinct positive primes. The manuscript explicitly corrects the four points that had prevented the previous version from being regarded as closed: (i) the notation for the fixed prime \(q₀\) is now kept uniformly distinct from the notation for the sieve moduli \(d\); (ii) the weight convention is unified by working with the function \(\vt(n)=log n\) on the primes and \(0\) off the primes, while \(Λ\) is used only inside the analytic estimates where it is the natural variable; (iii) the full residual notation \((W,a_W,b_W,S₁,A_d,g(d))\) has been incorporated throughout the manuscript; and (iv) the final closure is replaced by a residual-completion theorem on the common support of the core, thereby eliminating the logical gap produced by intersecting two independent theorems.

Citations

• Primes in arithmetic progressions to large moduli III: Uniform residue classes
• Primes in arithmetic progressions to large moduli II: Well-factorable estimates
• Binary Goldbach under EH(θ) and a Thin-Set BDH Mean-Square via a Hybrid Minor-Arc Method
• On systems of complexity one in the primes
• Linear Equations in Primes
• Primes in arithmetic progressions to large moduli

Discussions

• Some days ago this was published on ArXiv. It’s the outcome of many years of work, and it looks like we found something... arxiv.org/abs/2604.09753 [bsky, 1 points, 0 comments]