On the existence of magic squares of powers Rome, Nick Yamagishi, Shuntaro ddc:510 For any d 2, we prove that there exists an integer n0(d) such that there exists an n × n magic square of dth powers for all n n0(d). In particular, we establish the existence of an n × n magic square of squares for all n 4, which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of n × n magic squares of dth powers had only been known for sparse values of n. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity. Springer Nature 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/20423 https://research-explorer.ista.ac.at/download/20423/20463 Rome N, Yamagishi S. On the existence of magic squares of powers. <i>Research in Number Theory</i>. 2025;11(4). doi:<a href="https://doi.org/10.1007/s40993-025-00671-5">10.1007/s40993-025-00671-5</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s40993-025-00671-5 info:eu-repo/semantics/altIdentifier/e-issn/2363-9555 info:eu-repo/semantics/altIdentifier/arxiv/2406.09364 info:eu-repo/semantics/openAccess