[{"external_id":{"arxiv":["2406.09364"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"TiBr"}],"file_date_updated":"2025-10-13T11:28:49Z","PlanS_conform":"1","file":[{"date_updated":"2025-10-13T11:28:49Z","success":1,"file_id":"20463","checksum":"d41fbdc0cfc1fbceb519eb49b20a3ec2","date_created":"2025-10-13T11:28:49Z","content_type":"application/pdf","creator":"dernst","access_level":"open_access","file_name":"2025_ResearchNumberTheory_Rome.pdf","relation":"main_file","file_size":428531}],"issue":"4","publication_identifier":{"eissn":["2363-9555"]},"has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"title":"On the existence of magic squares of powers","acknowledgement":"The authors are grateful to Tim Browning for his constant encouragement and enthusiasm, Jörg Brüdern for very helpful discussion regarding his paper [1] and Diyuan Wu for turning the proof of Theorem 2.4 in the original version into an algorithm and running the computation for us, for which the results are available in the appendix of the original version. They would also like to thank Christian Boyer for maintaining his website [4] which contains a comprehensive list of various magic squares discovered, Brady Haran and Tony Várilly-Alvarado for their public engagement activity of mathematics and magic squares of squares (A YouTube video “Magic Squares of Squares (are PROBABLY impossible)” of the Numberphile channel by Brady Haran, in which Tony Várilly-Alvarado appears as a guest speaker: https://www.youtube.com/watch?v=Kdsj84UdeYg.), and all the magic squares enthusiasts who have contributed to [4] which made this paper possible. Finally, the authors would like to thank the anonymous referees for their helpful comments, Daniel Flores for his work [11] which inspired them to optimise the proof of Theorem 2.4 and Trevor Wooley for very helpful discussion regarding recent developments in Waring’s problem and his comments on the original version of this paper.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). NR was supported by FWF project ESP 441-NBL while SY by a FWF grant (DOI 10.55776/P32428).","intvolume":"        11","date_updated":"2025-10-13T12:30:40Z","type":"journal_article","date_published":"2025-09-23T00:00:00Z","date_created":"2025-10-05T22:01:34Z","year":"2025","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"OA_type":"hybrid","citation":{"apa":"Rome, N., &#38; Yamagishi, S. (2025). On the existence of magic squares of powers. <i>Research in Number Theory</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40993-025-00671-5\">https://doi.org/10.1007/s40993-025-00671-5</a>","chicago":"Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares of Powers.” <i>Research in Number Theory</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s40993-025-00671-5\">https://doi.org/10.1007/s40993-025-00671-5</a>.","ama":"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>","mla":"Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares of Powers.” <i>Research in Number Theory</i>, vol. 11, no. 4, 91, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s40993-025-00671-5\">10.1007/s40993-025-00671-5</a>.","ista":"Rome N, Yamagishi S. 2025. On the existence of magic squares of powers. Research in Number Theory. 11(4), 91.","ieee":"N. Rome and S. Yamagishi, “On the existence of magic squares of powers,” <i>Research in Number Theory</i>, vol. 11, no. 4. Springer Nature, 2025.","short":"N. Rome, S. Yamagishi, Research in Number Theory 11 (2025)."},"publication_status":"published","_id":"20423","publisher":"Springer Nature","publication":"Research in Number Theory","day":"23","oa":1,"arxiv":1,"author":[{"full_name":"Rome, Nick","last_name":"Rome","first_name":"Nick"},{"full_name":"Yamagishi, Shuntaro","last_name":"Yamagishi","id":"0c3fbc5c-f7a6-11ec-8d70-9485e75b416b","first_name":"Shuntaro"}],"doi":"10.1007/s40993-025-00671-5","month":"09","project":[{"name":"New frontiers of the Manin conjecture","call_identifier":"FWF","_id":"26AEDAB2-B435-11E9-9278-68D0E5697425","grant_number":"P32428"}],"volume":11,"scopus_import":"1","article_type":"original","oa_version":"Published Version","corr_author":"1","ddc":["510"],"quality_controlled":"1","OA_place":"publisher","abstract":[{"text":"For any d  2, we prove that there exists an integer n0(d) such that there exists an n × n\r\nmagic square of dth powers for all n  n0(d). In particular, we establish the existence of\r\nan n × n magic square of squares for all n  4, which settles a conjecture of\r\nVárilly-Alvarado. All previous approaches had been based on constructive methods and\r\nthe existence of n × n magic squares of dth powers had only been known for sparse\r\nvalues of n. We prove our result by the Hardy-Littlewood circle method, which in this\r\nsetting essentially reduces the problem to finding a sufficient number of disjoint linearly\r\nindependent subsets of the columns of the coefficient matrix of the equations defining\r\nmagic squares. We prove an optimal (up to a constant) lower bound for this quantity.","lang":"eng"}],"article_number":"91","status":"public"},{"external_id":{"arxiv":["2005.10188"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"title":"A density of ramified primes","publication_identifier":{"issn":["2522-0160"],"eissn":["2363-9555"]},"date_published":"2021-11-15T00:00:00Z","date_created":"2025-04-05T10:50:51Z","year":"2021","OA_type":"hybrid","citation":{"mla":"Chan, Stephanie, et al. “A Density of Ramified Primes.” <i>Research in Number Theory</i>, vol. 8, 1, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s40993-021-00295-5\">10.1007/s40993-021-00295-5</a>.","ista":"Chan S, McMeekin C, Milovic D. 2021. A density of ramified primes. Research in Number Theory. 8, 1.","apa":"Chan, S., McMeekin, C., &#38; Milovic, D. (2021). A density of ramified primes. <i>Research in Number Theory</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40993-021-00295-5\">https://doi.org/10.1007/s40993-021-00295-5</a>","chicago":"Chan, Stephanie, Christine McMeekin, and Djordjo Milovic. “A Density of Ramified Primes.” <i>Research in Number Theory</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s40993-021-00295-5\">https://doi.org/10.1007/s40993-021-00295-5</a>.","ama":"Chan S, McMeekin C, Milovic D. A density of ramified primes. <i>Research in Number Theory</i>. 2021;8. doi:<a href=\"https://doi.org/10.1007/s40993-021-00295-5\">10.1007/s40993-021-00295-5</a>","ieee":"S. Chan, C. McMeekin, and D. Milovic, “A density of ramified primes,” <i>Research in Number Theory</i>, vol. 8. Springer Nature, 2021.","short":"S. Chan, C. McMeekin, D. Milovic, Research in Number Theory 8 (2021)."},"language":[{"iso":"eng"}],"article_processing_charge":"No","publication_status":"published","date_updated":"2025-07-10T11:51:46Z","intvolume":"         8","type":"journal_article","_id":"19489","publication":"Research in Number Theory","oa":1,"day":"15","arxiv":1,"publisher":"Springer Nature","scopus_import":"1","volume":8,"article_type":"original","oa_version":"Published Version","author":[{"last_name":"Chan","first_name":"Yik Tung","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung"},{"first_name":"Christine","last_name":"McMeekin","full_name":"McMeekin, Christine"},{"last_name":"Milovic","first_name":"Djordjo","full_name":"Milovic, Djordjo"}],"doi":"10.1007/s40993-021-00295-5","month":"11","OA_place":"publisher","ddc":["510"],"quality_controlled":"1","status":"public","extern":"1","main_file_link":[{"url":"https://doi.org/10.1007/s40993-021-00295-5","open_access":"1"}],"article_number":"1","abstract":[{"lang":"eng","text":"Let K be a cyclic number field of odd degree over \r\n𝑄 with odd narrow class number, such that 2 is inert in 𝐾/𝑄. We define a family of number fields {𝐾(𝑝)}𝑝, depending on K and indexed by the rational primes p that split completely in 𝐾/𝑄, in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in 𝐾(𝑝)/𝑄 is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension 𝐾/𝑄. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals."}]},{"publication_identifier":{"issn":["2522-0160"],"eissn":["2363-9555"]},"title":"Modular invariants for genus 3 hyperelliptic curves","department":[{"_id":"TiBr"}],"external_id":{"arxiv":["1807.08986"]},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"10874","intvolume":"         5","date_updated":"2023-09-05T15:39:31Z","type":"journal_article","acknowledgement":"The authors would like to thank the Lorentz Center in Leiden for hosting the Women in Numbers Europe 2 workshop and providing a productive and enjoyable environment for our initial work on this project. We are grateful to the organizers of WIN-E2, Irene Bouw, Rachel Newton and Ekin Ozman, for making this conference and this collaboration possible. We\r\nthank Irene Bouw and Christophe Ritzenhaler for helpful discussions. Ionica acknowledges support from the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. Most of Kılıçer’s work was carried out during her stay in Universiteit Leiden and Carl von Ossietzky Universität Oldenburg. Massierer was supported by the Australian Research Council (DP150101689). Vincent is supported by the National Science Foundation under Grant No. DMS-1802323 and by the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. ","citation":{"ieee":"S. Ionica <i>et al.</i>, “Modular invariants for genus 3 hyperelliptic curves,” <i>Research in Number Theory</i>, vol. 5. Springer Nature, 2019.","short":"S. Ionica, P. Kılıçer, K. Lauter, E. Lorenzo García, M.-A. Manzateanu, M. Massierer, C. Vincent, Research in Number Theory 5 (2019).","ama":"Ionica S, Kılıçer P, Lauter K, et al. Modular invariants for genus 3 hyperelliptic curves. <i>Research in Number Theory</i>. 2019;5. doi:<a href=\"https://doi.org/10.1007/s40993-018-0146-6\">10.1007/s40993-018-0146-6</a>","chicago":"Ionica, Sorina, Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Maria-Adelina Manzateanu, Maike Massierer, and Christelle Vincent. “Modular Invariants for Genus 3 Hyperelliptic Curves.” <i>Research in Number Theory</i>. Springer Nature, 2019. <a href=\"https://doi.org/10.1007/s40993-018-0146-6\">https://doi.org/10.1007/s40993-018-0146-6</a>.","apa":"Ionica, S., Kılıçer, P., Lauter, K., Lorenzo García, E., Manzateanu, M.-A., Massierer, M., &#38; Vincent, C. (2019). Modular invariants for genus 3 hyperelliptic curves. <i>Research in Number Theory</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40993-018-0146-6\">https://doi.org/10.1007/s40993-018-0146-6</a>","ista":"Ionica S, Kılıçer P, Lauter K, Lorenzo García E, Manzateanu M-A, Massierer M, Vincent C. 2019. Modular invariants for genus 3 hyperelliptic curves. Research in Number Theory. 5, 9.","mla":"Ionica, Sorina, et al. “Modular Invariants for Genus 3 Hyperelliptic Curves.” <i>Research in Number Theory</i>, vol. 5, 9, Springer Nature, 2019, doi:<a href=\"https://doi.org/10.1007/s40993-018-0146-6\">10.1007/s40993-018-0146-6</a>."},"language":[{"iso":"eng"}],"article_processing_charge":"No","publication_status":"published","date_published":"2019-01-02T00:00:00Z","date_created":"2022-03-18T12:09:48Z","year":"2019","month":"01","author":[{"last_name":"Ionica","first_name":"Sorina","full_name":"Ionica, Sorina"},{"full_name":"Kılıçer, Pınar","last_name":"Kılıçer","first_name":"Pınar"},{"full_name":"Lauter, Kristin","last_name":"Lauter","first_name":"Kristin"},{"full_name":"Lorenzo García, Elisa","first_name":"Elisa","last_name":"Lorenzo García"},{"full_name":"Manzateanu, Maria-Adelina","last_name":"Manzateanu","first_name":"Maria-Adelina","id":"be8d652e-a908-11ec-82a4-e2867729459c"},{"full_name":"Massierer, Maike","last_name":"Massierer","first_name":"Maike"},{"full_name":"Vincent, Christelle","first_name":"Christelle","last_name":"Vincent"}],"doi":"10.1007/s40993-018-0146-6","oa_version":"Preprint","volume":5,"scopus_import":"1","article_type":"original","publisher":"Springer Nature","arxiv":1,"publication":"Research in Number Theory","day":"02","oa":1,"abstract":[{"text":"In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.","lang":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/1807.08986","open_access":"1"}],"article_number":"9","keyword":["Algebra and Number Theory"],"status":"public","quality_controlled":"1"}]