---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20423'
abstract:
- lang: eng
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."
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)."
article_number: '91'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Nick
full_name: Rome, Nick
last_name: Rome
- first_name: Shuntaro
full_name: Yamagishi, Shuntaro
id: 0c3fbc5c-f7a6-11ec-8d70-9485e75b416b
last_name: Yamagishi
citation:
ama: Rome N, Yamagishi S. On the existence of magic squares of powers. Research
in Number Theory. 2025;11(4). doi:10.1007/s40993-025-00671-5
apa: Rome, N., & Yamagishi, S. (2025). On the existence of magic squares of
powers. Research in Number Theory. Springer Nature. https://doi.org/10.1007/s40993-025-00671-5
chicago: Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares
of Powers.” Research in Number Theory. Springer Nature, 2025. https://doi.org/10.1007/s40993-025-00671-5.
ieee: N. Rome and S. Yamagishi, “On the existence of magic squares of powers,” Research
in Number Theory, vol. 11, no. 4. Springer Nature, 2025.
ista: Rome N, Yamagishi S. 2025. On the existence of magic squares of powers. Research
in Number Theory. 11(4), 91.
mla: Rome, Nick, and Shuntaro Yamagishi. “On the Existence of Magic Squares of Powers.”
Research in Number Theory, vol. 11, no. 4, 91, Springer Nature, 2025, doi:10.1007/s40993-025-00671-5.
short: N. Rome, S. Yamagishi, Research in Number Theory 11 (2025).
corr_author: '1'
date_created: 2025-10-05T22:01:34Z
date_published: 2025-09-23T00:00:00Z
date_updated: 2025-10-13T12:30:40Z
day: '23'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s40993-025-00671-5
external_id:
arxiv:
- '2406.09364'
file:
- access_level: open_access
checksum: d41fbdc0cfc1fbceb519eb49b20a3ec2
content_type: application/pdf
creator: dernst
date_created: 2025-10-13T11:28:49Z
date_updated: 2025-10-13T11:28:49Z
file_id: '20463'
file_name: 2025_ResearchNumberTheory_Rome.pdf
file_size: 428531
relation: main_file
success: 1
file_date_updated: 2025-10-13T11:28:49Z
has_accepted_license: '1'
intvolume: ' 11'
issue: '4'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P32428
name: New frontiers of the Manin conjecture
publication: Research in Number Theory
publication_identifier:
eissn:
- 2363-9555
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the existence of magic squares of powers
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 11
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '19489'
abstract:
- lang: eng
text: "Let K be a cyclic number field of odd degree over \r\n\U0001D444 with odd
narrow class number, such that 2 is inert in \U0001D43E/\U0001D444. We define
a family of number fields {\U0001D43E(\U0001D45D)}\U0001D45D, depending on K and
indexed by the rational primes p that split completely in \U0001D43E/\U0001D444,
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 \U0001D43E(\U0001D45D)/\U0001D444 is
strictly between 0 and 1 and is given explicitly as a formula in terms of the
degree of the extension \U0001D43E/\U0001D444. Our results are unconditional in
the cubic case. Our proof relies on a detailed study of the joint distribution
of spins of prime ideals."
article_number: '1'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yik Tung
full_name: Chan, Yik Tung
id: c4c0afc8-9262-11ed-9231-d8b0bc743af1
last_name: Chan
orcid: 0000-0001-8467-4106
- first_name: Christine
full_name: McMeekin, Christine
last_name: McMeekin
- first_name: Djordjo
full_name: Milovic, Djordjo
last_name: Milovic
citation:
ama: Chan S, McMeekin C, Milovic D. A density of ramified primes. Research in
Number Theory. 2021;8. doi:10.1007/s40993-021-00295-5
apa: Chan, S., McMeekin, C., & Milovic, D. (2021). A density of ramified primes.
Research in Number Theory. Springer Nature. https://doi.org/10.1007/s40993-021-00295-5
chicago: Chan, Stephanie, Christine McMeekin, and Djordjo Milovic. “A Density of
Ramified Primes.” Research in Number Theory. Springer Nature, 2021. https://doi.org/10.1007/s40993-021-00295-5.
ieee: S. Chan, C. McMeekin, and D. Milovic, “A density of ramified primes,” Research
in Number Theory, vol. 8. Springer Nature, 2021.
ista: Chan S, McMeekin C, Milovic D. 2021. A density of ramified primes. Research
in Number Theory. 8, 1.
mla: Chan, Stephanie, et al. “A Density of Ramified Primes.” Research in Number
Theory, vol. 8, 1, Springer Nature, 2021, doi:10.1007/s40993-021-00295-5.
short: S. Chan, C. McMeekin, D. Milovic, Research in Number Theory 8 (2021).
date_created: 2025-04-05T10:50:51Z
date_published: 2021-11-15T00:00:00Z
date_updated: 2025-07-10T11:51:46Z
day: '15'
ddc:
- '510'
doi: 10.1007/s40993-021-00295-5
extern: '1'
external_id:
arxiv:
- '2005.10188'
has_accepted_license: '1'
intvolume: ' 8'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s40993-021-00295-5
month: '11'
oa: 1
oa_version: Published Version
publication: Research in Number Theory
publication_identifier:
eissn:
- 2363-9555
issn:
- 2522-0160
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: A density of ramified primes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2021'
...
---
_id: '10874'
abstract:
- lang: eng
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.
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. "
article_number: '9'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sorina
full_name: Ionica, Sorina
last_name: Ionica
- first_name: Pınar
full_name: Kılıçer, Pınar
last_name: Kılıçer
- first_name: Kristin
full_name: Lauter, Kristin
last_name: Lauter
- first_name: Elisa
full_name: Lorenzo García, Elisa
last_name: Lorenzo García
- first_name: Maria-Adelina
full_name: Manzateanu, Maria-Adelina
id: be8d652e-a908-11ec-82a4-e2867729459c
last_name: Manzateanu
- first_name: Maike
full_name: Massierer, Maike
last_name: Massierer
- first_name: Christelle
full_name: Vincent, Christelle
last_name: Vincent
citation:
ama: Ionica S, Kılıçer P, Lauter K, et al. Modular invariants for genus 3 hyperelliptic
curves. Research in Number Theory. 2019;5. doi:10.1007/s40993-018-0146-6
apa: 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. Springer Nature. https://doi.org/10.1007/s40993-018-0146-6
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.” Research in Number Theory. Springer Nature, 2019.
https://doi.org/10.1007/s40993-018-0146-6.
ieee: S. Ionica et al., “Modular invariants for genus 3 hyperelliptic curves,”
Research in Number Theory, vol. 5. Springer Nature, 2019.
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.”
Research in Number Theory, vol. 5, 9, Springer Nature, 2019, doi:10.1007/s40993-018-0146-6.
short: S. Ionica, P. Kılıçer, K. Lauter, E. Lorenzo García, M.-A. Manzateanu, M.
Massierer, C. Vincent, Research in Number Theory 5 (2019).
date_created: 2022-03-18T12:09:48Z
date_published: 2019-01-02T00:00:00Z
date_updated: 2023-09-05T15:39:31Z
day: '02'
department:
- _id: TiBr
doi: 10.1007/s40993-018-0146-6
external_id:
arxiv:
- '1807.08986'
intvolume: ' 5'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1807.08986
month: '01'
oa: 1
oa_version: Preprint
publication: Research in Number Theory
publication_identifier:
eissn:
- 2363-9555
issn:
- 2522-0160
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Modular invariants for genus 3 hyperelliptic curves
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 5
year: '2019'
...