Pick any prime number larger than five, and it is guaranteed to fit into a perfect, all-prime magic square.
April 16, 2026
Original Paper
Universal Inclusion of Prescribed Primes in 3x3 Magic Squares
arXiv · 2604.09753
The Takeaway
Mathematicians have been obsessed with magic squares—grids where every row, column, and diagonal adds up to the same number—for thousands of years. For a long time, finding one made entirely of prime numbers felt like searching for a needle in a haystack, and no one knew if every prime could actually participate in one. This paper finally proves that prime numbers aren't just random; they possess a deep, structural flexibility that allows every single prime from 5 to infinity to be part of a distinct 3x3 magic square. It reveals a hidden symmetry in the distribution of primes that we previously only guessed at. For us, it’s a reminder that even the most chaotic-seeming parts of our universe are governed by a weirdly beautiful, rigid order.
From the abstract
We present an integrated version of the global program proving that every prescribed prime \(q_0\ge 5\) occurs in some \(3\times 3\) 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 \(q_0\) is now kept uniformly distinct from the notation for the sieve moduli \(d\); (ii) the weight convention is unified by working with the fun