[{"year":"2025","acknowledgement":"While working on this paper, the first author was supported by a FWF grant (DOI 10.55776/P36278).","oa":1,"page":"1677-1691","author":[{"orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D","last_name":"Browning","first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87"},{"id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","first_name":"Yik Tung","last_name":"Chan","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"text":"The large sieve is used to estimate the density of quadratic polynomials Q ∈ Z[x],\r\nsuch that there exists an odd degree polynomial defined over Z which has resultant ±1 with Q.\r\nGiven a monic polynomial R ∈ Z[x] of odd degree, this is used to show that for almost all\r\nquadratic polynomials Q ∈ Z[x], there exists a prime p such that Q and R share a common\r\nroot in Fp. Using recent work of Landesman, an application to the average size of the odd part\r\nof the class group of quadratic number fields is also given","lang":"eng"},{"text":" Le grand crible est utilisé pour estimer la densité des polynômes quadratiques Q ∈ Z[x] tels qu’il existe un polynôme de degré impair défini sur Z dont le résultant avec Q est égal à ±1. Étant donné un polynôme unitaire R ∈ Z[x] de degré impair, on s’en sert pour montrer que, pour presque tous les polynômes quadratiques Q ∈ Z[x], il existe un nombre premier p tel que Q et R aient une racine commune dans Fp. En utilisant des travaux récents de Landesman, on obtient également une application concernant la taille moyenne de la partie impaire du groupe de classe des corps quadratiques.","lang":"fre"}],"article_processing_charge":"Yes","publication_identifier":{"eissn":["2270-518X"],"issn":["2429-7100"]},"date_published":"2025-10-21T00:00:00Z","_id":"21343","has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"10","publication_status":"published","oa_version":"Published Version","type":"journal_article","quality_controlled":"1","external_id":{"arxiv":["2411.09264"]},"intvolume":"        12","publication":"Journal de l'ecole polytechnique mathematiques","DOAJ_listed":"1","file_date_updated":"2026-02-24T07:56:34Z","title":"Solubility of a resultant equation and applications","date_updated":"2026-02-24T07:57:53Z","doi":"10.5802/jep.320","corr_author":"1","file":[{"date_updated":"2026-02-24T07:56:34Z","content_type":"application/pdf","access_level":"open_access","success":1,"creator":"dernst","file_name":"2025_JEP_Browning.pdf","file_size":1003689,"file_id":"21356","date_created":"2026-02-24T07:56:34Z","relation":"main_file","checksum":"828577ea48ac6109d3e9dd1aeddd45c4"}],"OA_place":"publisher","department":[{"_id":"TiBr"}],"project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory","grant_number":"P36278"}],"citation":{"chicago":"Browning, Timothy D, and Stephanie Chan. “Solubility of a Resultant Equation and Applications.” <i>Journal de l’ecole Polytechnique Mathematiques</i>. Ecole polytechnique, 2025. <a href=\"https://doi.org/10.5802/jep.320\">https://doi.org/10.5802/jep.320</a>.","ista":"Browning TD, Chan S. 2025. Solubility of a resultant equation and applications. Journal de l’ecole polytechnique mathematiques. 12, 1677–1691.","short":"T.D. Browning, S. Chan, Journal de l’ecole Polytechnique Mathematiques 12 (2025) 1677–1691.","ieee":"T. D. Browning and S. Chan, “Solubility of a resultant equation and applications,” <i>Journal de l’ecole polytechnique mathematiques</i>, vol. 12. Ecole polytechnique, pp. 1677–1691, 2025.","apa":"Browning, T. D., &#38; Chan, S. (2025). Solubility of a resultant equation and applications. <i>Journal de l’ecole Polytechnique Mathematiques</i>. Ecole polytechnique. <a href=\"https://doi.org/10.5802/jep.320\">https://doi.org/10.5802/jep.320</a>","ama":"Browning TD, Chan S. Solubility of a resultant equation and applications. <i>Journal de l’ecole polytechnique mathematiques</i>. 2025;12:1677-1691. doi:<a href=\"https://doi.org/10.5802/jep.320\">10.5802/jep.320</a>","mla":"Browning, Timothy D., and Stephanie Chan. “Solubility of a Resultant Equation and Applications.” <i>Journal de l’ecole Polytechnique Mathematiques</i>, vol. 12, Ecole polytechnique, 2025, pp. 1677–91, doi:<a href=\"https://doi.org/10.5802/jep.320\">10.5802/jep.320</a>."},"article_type":"original","ddc":["510"],"language":[{"iso":"eng"}],"day":"21","scopus_import":"1","status":"public","OA_type":"gold","arxiv":1,"PlanS_conform":"1","date_created":"2026-02-22T23:01:36Z","publisher":"Ecole polytechnique","volume":12},{"file":[{"content_type":"application/pdf","creator":"dernst","access_level":"open_access","success":1,"date_updated":"2025-12-30T08:05:42Z","checksum":"752c407eb186d391380b10a7505f66cf","date_created":"2025-12-30T08:05:42Z","relation":"main_file","file_name":"2025_JourNumberTheory_Chan.pdf","file_size":685204,"file_id":"20889"}],"OA_place":"publisher","corr_author":"1","citation":{"ama":"Chan S, Koymans P, Pagano C, Sofos E. Averages of multiplicative functions along equidistributed sequences. <i>Journal of Number Theory</i>. 2025;273:1-36. doi:<a href=\"https://doi.org/10.1016/j.jnt.2025.01.005\">10.1016/j.jnt.2025.01.005</a>","mla":"Chan, Stephanie, et al. “Averages of Multiplicative Functions along Equidistributed Sequences.” <i>Journal of Number Theory</i>, vol. 273, Elsevier, 2025, pp. 1–36, doi:<a href=\"https://doi.org/10.1016/j.jnt.2025.01.005\">10.1016/j.jnt.2025.01.005</a>.","apa":"Chan, S., Koymans, P., Pagano, C., &#38; Sofos, E. (2025). Averages of multiplicative functions along equidistributed sequences. <i>Journal of Number Theory</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jnt.2025.01.005\">https://doi.org/10.1016/j.jnt.2025.01.005</a>","ista":"Chan S, Koymans P, Pagano C, Sofos E. 2025. Averages of multiplicative functions along equidistributed sequences. Journal of Number Theory. 273, 1–36.","chicago":"Chan, Stephanie, Peter Koymans, Carlo Pagano, and Efthymios Sofos. “Averages of Multiplicative Functions along Equidistributed Sequences.” <i>Journal of Number Theory</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.jnt.2025.01.005\">https://doi.org/10.1016/j.jnt.2025.01.005</a>.","short":"S. Chan, P. Koymans, C. Pagano, E. Sofos, Journal of Number Theory 273 (2025) 1–36.","ieee":"S. Chan, P. Koymans, C. Pagano, and E. Sofos, “Averages of multiplicative functions along equidistributed sequences,” <i>Journal of Number Theory</i>, vol. 273. Elsevier, pp. 1–36, 2025."},"department":[{"_id":"TiBr"}],"file_date_updated":"2025-12-30T08:05:42Z","publication":"Journal of Number Theory","intvolume":"       273","external_id":{"isi":["001444208500001"]},"doi":"10.1016/j.jnt.2025.01.005","date_updated":"2025-12-30T08:06:16Z","title":"Averages of multiplicative functions along equidistributed sequences","PlanS_conform":"1","date_created":"2025-03-09T23:01:26Z","OA_type":"hybrid","status":"public","volume":273,"publisher":"Elsevier","day":"01","language":[{"iso":"eng"}],"ddc":["510"],"article_type":"original","scopus_import":"1","article_processing_charge":"Yes (in subscription journal)","isi":1,"publication_identifier":{"issn":["0022-314X"]},"oa":1,"year":"2025","abstract":[{"text":"For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.","lang":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"author":[{"first_name":"Yik Tung","last_name":"Chan","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung"},{"first_name":"Peter","last_name":"Koymans","full_name":"Koymans, Peter"},{"full_name":"Pagano, Carlo","first_name":"Carlo","last_name":"Pagano"},{"last_name":"Sofos","first_name":"Efthymios","full_name":"Sofos, Efthymios"}],"page":"1-36","oa_version":"Published Version","publication_status":"published","month":"08","quality_controlled":"1","type":"journal_article","_id":"19363","has_accepted_license":"1","date_published":"2025-08-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"date_created":"2025-04-05T10:49:27Z","status":"public","OA_type":"green","arxiv":1,"publisher":"Scuola Normale Superiore - Edizioni della Normale","article_number":"18","day":"07","language":[{"iso":"eng"}],"article_type":"original","OA_place":"repository","corr_author":"1","citation":{"ama":"Chan S, Koymans P, Pagano C, Sofos E. 6-torision and integral points on quartic threefolds. <i>Annali della Scuola Normale Superiore di Pisa, Classe di Scienze</i>. 2025. doi:<a href=\"https://doi.org/10.2422/2036-2145.202412_006\">10.2422/2036-2145.202412_006</a>","mla":"Chan, Stephanie, et al. “6-Torision and Integral Points on Quartic Threefolds.” <i>Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze</i>, 18, Scuola Normale Superiore - Edizioni della Normale, 2025, doi:<a href=\"https://doi.org/10.2422/2036-2145.202412_006\">10.2422/2036-2145.202412_006</a>.","apa":"Chan, S., Koymans, P., Pagano, C., &#38; Sofos, E. (2025). 6-torision and integral points on quartic threefolds. <i>Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze</i>. Scuola Normale Superiore - Edizioni della Normale. <a href=\"https://doi.org/10.2422/2036-2145.202412_006\">https://doi.org/10.2422/2036-2145.202412_006</a>","short":"S. Chan, P. Koymans, C. Pagano, E. Sofos, Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze (2025).","ista":"Chan S, Koymans P, Pagano C, Sofos E. 2025. 6-torision and integral points on quartic threefolds. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze., 18.","chicago":"Chan, Stephanie, Peter Koymans, Carlo Pagano, and Efthymios Sofos. “6-Torision and Integral Points on Quartic Threefolds.” <i>Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze</i>. Scuola Normale Superiore - Edizioni della Normale, 2025. <a href=\"https://doi.org/10.2422/2036-2145.202412_006\">https://doi.org/10.2422/2036-2145.202412_006</a>.","ieee":"S. Chan, P. Koymans, C. Pagano, and E. Sofos, “6-torision and integral points on quartic threefolds,” <i>Annali della Scuola Normale Superiore di Pisa, Classe di Scienze</i>. Scuola Normale Superiore - Edizioni della Normale, 2025."},"department":[{"_id":"TiBr"}],"publication":"Annali della Scuola Normale Superiore di Pisa, Classe di Scienze","external_id":{"arxiv":["2403.13359"]},"title":"6-torision and integral points on quartic threefolds","date_updated":"2025-05-14T11:40:24Z","doi":"10.2422/2036-2145.202412_006","publication_status":"epub_ahead","month":"03","oa_version":"Preprint","type":"journal_article","date_published":"2025-03-07T00:00:00Z","_id":"19483","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2403.13359"}],"publication_identifier":{"eissn":["2036-2145"],"issn":["0391-173X"]},"oa":1,"year":"2025","abstract":[{"text":"We prove matching upper and lower bounds for the average of the6-torsionof class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.","lang":"eng"}],"author":[{"first_name":"Yik Tung","last_name":"Chan","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","full_name":"Chan, Yik Tung","orcid":"0000-0001-8467-4106"},{"first_name":"Peter","last_name":"Koymans","full_name":"Koymans, Peter"},{"last_name":"Pagano","first_name":"Carlo","full_name":"Pagano, Carlo"},{"full_name":"Sofos, Efthymios","last_name":"Sofos","first_name":"Efthymios"}]},{"publication":"Journal of the European Mathematical Society","DOAJ_listed":"1","das_tickbox":"1","external_id":{"arxiv":["2401.04375"]},"doi":"10.4171/jems/1704","date_updated":"2026-07-06T11:51:48Z","title":"Almost all quadratic twists of an elliptic curve have no integral points","OA_place":"publisher","corr_author":"1","citation":{"apa":"Browning, T. D., &#38; Chan, S. (2025). Almost all quadratic twists of an elliptic curve have no integral points. <i>Journal of the European Mathematical Society</i>. EMS Press. <a href=\"https://doi.org/10.4171/jems/1704\">https://doi.org/10.4171/jems/1704</a>","mla":"Browning, Timothy D., and Stephanie Chan. “Almost All Quadratic Twists of an Elliptic Curve Have No Integral Points.” <i>Journal of the European Mathematical Society</i>, EMS Press, 2025, doi:<a href=\"https://doi.org/10.4171/jems/1704\">10.4171/jems/1704</a>.","ama":"Browning TD, Chan S. Almost all quadratic twists of an elliptic curve have no integral points. <i>Journal of the European Mathematical Society</i>. 2025. doi:<a href=\"https://doi.org/10.4171/jems/1704\">10.4171/jems/1704</a>","ieee":"T. D. Browning and S. Chan, “Almost all quadratic twists of an elliptic curve have no integral points,” <i>Journal of the European Mathematical Society</i>. EMS Press, 2025.","ista":"Browning TD, Chan S. 2025. Almost all quadratic twists of an elliptic curve have no integral points. Journal of the European Mathematical Society.","chicago":"Browning, Timothy D, and Stephanie Chan. “Almost All Quadratic Twists of an Elliptic Curve Have No Integral Points.” <i>Journal of the European Mathematical Society</i>. EMS Press, 2025. <a href=\"https://doi.org/10.4171/jems/1704\">https://doi.org/10.4171/jems/1704</a>.","short":"T.D. Browning, S. Chan, Journal of the European Mathematical Society (2025)."},"project":[{"grant_number":"P36278","name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3"}],"department":[{"_id":"TiBr"}],"language":[{"iso":"eng"}],"day":"17","ddc":["510"],"article_type":"original","date_created":"2026-02-17T07:46:26Z","OA_type":"diamond","arxiv":1,"status":"public","publisher":"EMS Press","oa":1,"acknowledgement":"The authors are grateful to Roger Heath-Brown and to the anonymous referees for useful comments. The first author was supported by an FWF grant (DOI 10.55776/P36278).","year":"2025","abstract":[{"text":"For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E \r\nD have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall–Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown’s analysis of Selmer group statistics for the congruent number curve.","lang":"eng"}],"author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177"},{"full_name":"Chan, Yik Tung","orcid":"0000-0001-8467-4106","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","last_name":"Chan","first_name":"Yik Tung"}],"article_processing_charge":"No","main_file_link":[{"open_access":"1","url":"https://doi.org/10.4171/JEMS/1704"}],"publication_identifier":{"issn":["1435-9855"],"eissn":["1435-9863"]},"_id":"21266","date_published":"2025-09-17T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","publication_status":"epub_ahead","month":"09","quality_controlled":"1","type":"journal_article"},{"article_processing_charge":"No","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2211.06062"}],"publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"oa":1,"acknowledgement":"The author would like to thank Peter Koymans and Carlo Pagano for helpful discussions.","year":"2024","abstract":[{"text":"Consider the family of elliptic curves En:y2=x3+n2, where n varies over positive cubefree integers. There is a rational 3-isogeny ϕ from En to E^n:y2=x3−27n2 and a dual isogeny ϕ^:E^n→En. We show that for almost all n, the rank of Selϕ(En) is 0, and the rank of Selϕ^(E^n) is determined by the number of prime factors of n that are congruent to 2mod3 and the congruence class of nmod9.","lang":"eng"}],"issue":"9","author":[{"first_name":"Yik Tung","last_name":"Chan","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","full_name":"Chan, Yik Tung","orcid":"0000-0001-8467-4106"}],"page":"7571-7593","oa_version":"Preprint","publication_status":"published","month":"05","quality_controlled":"1","type":"journal_article","_id":"19486","date_published":"2024-05-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"repository","citation":{"ieee":"S. Chan, “The 3-isogeny selmer groups of the elliptic curves y2=x3+n2,” <i>International Mathematics Research Notices</i>, vol. 2024, no. 9. Oxford University Press, pp. 7571–7593, 2024.","short":"S. Chan, International Mathematics Research Notices 2024 (2024) 7571–7593.","ista":"Chan S. 2024. The 3-isogeny selmer groups of the elliptic curves y2=x3+n2. International Mathematics Research Notices. 2024(9), 7571–7593.","chicago":"Chan, Stephanie. “The 3-Isogeny Selmer Groups of the Elliptic Curves Y2=x3+n2.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2024. <a href=\"https://doi.org/10.1093/imrn/rnad266\">https://doi.org/10.1093/imrn/rnad266</a>.","apa":"Chan, S. (2024). The 3-isogeny selmer groups of the elliptic curves y2=x3+n2. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnad266\">https://doi.org/10.1093/imrn/rnad266</a>","mla":"Chan, Stephanie. “The 3-Isogeny Selmer Groups of the Elliptic Curves Y2=x3+n2.” <i>International Mathematics Research Notices</i>, vol. 2024, no. 9, Oxford University Press, 2024, pp. 7571–93, doi:<a href=\"https://doi.org/10.1093/imrn/rnad266\">10.1093/imrn/rnad266</a>.","ama":"Chan S. The 3-isogeny selmer groups of the elliptic curves y2=x3+n2. <i>International Mathematics Research Notices</i>. 2024;2024(9):7571-7593. doi:<a href=\"https://doi.org/10.1093/imrn/rnad266\">10.1093/imrn/rnad266</a>"},"publication":"International Mathematics Research Notices","intvolume":"      2024","external_id":{"arxiv":["2211.06062"]},"doi":"10.1093/imrn/rnad266","title":"The 3-isogeny selmer groups of the elliptic curves y2=x3+n2","date_updated":"2025-07-10T11:51:44Z","date_created":"2025-04-05T10:50:33Z","OA_type":"green","arxiv":1,"status":"public","volume":2024,"extern":"1","publisher":"Oxford University Press","day":"01","language":[{"iso":"eng"}],"article_type":"original","scopus_import":"1"},{"article_processing_charge":"Yes (via OA deal)","publication_identifier":{"issn":["0001-8708"],"eissn":["1090-2082"]},"year":"2024","oa":1,"author":[{"full_name":"Chan, Yik Tung","orcid":"0000-0001-8467-4106","first_name":"Yik Tung","last_name":"Chan","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"text":" We show that the total number of non-torsion integral points on the elliptic curves ED : y\r\n2 = x3 − D2x, where D ranges over positive squarefree integers less than N, is O(N(log N)\r\n−1/4+ǫ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown’s method on estimating the average size of the 2-Selmer group of the curves in this family.","lang":"eng"}],"issue":"11","month":"11","publication_status":"published","oa_version":"Published Version","type":"journal_article","quality_controlled":"1","date_published":"2024-11-01T00:00:00Z","_id":"18064","has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","corr_author":"1","file":[{"access_level":"open_access","creator":"dernst","success":1,"content_type":"application/pdf","date_updated":"2025-01-13T08:54:09Z","checksum":"f555742540ad91a3040aeafd68b1fcde","relation":"main_file","date_created":"2025-01-13T08:54:09Z","file_id":"18829","file_size":564386,"file_name":"2024_AdvancesMath_Chan.pdf"}],"OA_place":"publisher","department":[{"_id":"TiBr"}],"citation":{"apa":"Chan, S. (2024). The average number of integral points on the congruent number curves. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2024.109946\">https://doi.org/10.1016/j.aim.2024.109946</a>","mla":"Chan, Stephanie. “The Average Number of Integral Points on the Congruent Number Curves.” <i>Advances in Mathematics</i>, vol. 457, no. 11, 109946, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109946\">10.1016/j.aim.2024.109946</a>.","ama":"Chan S. The average number of integral points on the congruent number curves. <i>Advances in Mathematics</i>. 2024;457(11). doi:<a href=\"https://doi.org/10.1016/j.aim.2024.109946\">10.1016/j.aim.2024.109946</a>","ieee":"S. Chan, “The average number of integral points on the congruent number curves,” <i>Advances in Mathematics</i>, vol. 457, no. 11. Elsevier, 2024.","chicago":"Chan, Stephanie. “The Average Number of Integral Points on the Congruent Number Curves.” <i>Advances in Mathematics</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.aim.2024.109946\">https://doi.org/10.1016/j.aim.2024.109946</a>.","short":"S. Chan, Advances in Mathematics 457 (2024).","ista":"Chan S. 2024. The average number of integral points on the congruent number curves. Advances in Mathematics. 457(11), 109946."},"external_id":{"arxiv":["2112.01615"]},"intvolume":"       457","publication":"Advances in Mathematics","file_date_updated":"2025-01-13T08:54:09Z","date_updated":"2025-01-13T08:54:36Z","title":"The average number of integral points on the congruent number curves","doi":"10.1016/j.aim.2024.109946","status":"public","arxiv":1,"OA_type":"hybrid","date_created":"2024-09-15T22:01:39Z","publisher":"Elsevier","volume":457,"article_type":"original","ddc":["510"],"article_number":"109946","day":"01","language":[{"iso":"eng"}],"scopus_import":"1"},{"intvolume":"       388","external_id":{"arxiv":["2203.11366"]},"publication":"Mathematische Annalen","doi":"10.1007/s00208-023-02578-x","title":"Integral points on cubic twists of Mordell curves","date_updated":"2025-07-10T11:51:45Z","OA_place":"repository","citation":{"ista":"Chan S. 2023. Integral points on cubic twists of Mordell curves. Mathematische Annalen. 388(3), 2275–2288.","short":"S. Chan, Mathematische Annalen 388 (2023) 2275–2288.","chicago":"Chan, Stephanie. “Integral Points on Cubic Twists of Mordell Curves.” <i>Mathematische Annalen</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00208-023-02578-x\">https://doi.org/10.1007/s00208-023-02578-x</a>.","ieee":"S. Chan, “Integral points on cubic twists of Mordell curves,” <i>Mathematische Annalen</i>, vol. 388, no. 3. Springer Nature, pp. 2275–2288, 2023.","ama":"Chan S. Integral points on cubic twists of Mordell curves. <i>Mathematische Annalen</i>. 2023;388(3):2275-2288. doi:<a href=\"https://doi.org/10.1007/s00208-023-02578-x\">10.1007/s00208-023-02578-x</a>","mla":"Chan, Stephanie. “Integral Points on Cubic Twists of Mordell Curves.” <i>Mathematische Annalen</i>, vol. 388, no. 3, Springer Nature, 2023, pp. 2275–88, doi:<a href=\"https://doi.org/10.1007/s00208-023-02578-x\">10.1007/s00208-023-02578-x</a>.","apa":"Chan, S. (2023). Integral points on cubic twists of Mordell curves. <i>Mathematische Annalen</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00208-023-02578-x\">https://doi.org/10.1007/s00208-023-02578-x</a>"},"article_type":"original","day":"07","language":[{"iso":"eng"}],"scopus_import":"1","arxiv":1,"OA_type":"green","status":"public","date_created":"2025-04-05T10:50:37Z","extern":"1","publisher":"Springer Nature","volume":388,"year":"2023","oa":1,"author":[{"id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","last_name":"Chan","first_name":"Yik Tung","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung"}],"page":"2275-2288","issue":"3","abstract":[{"lang":"eng","text":"Fix a non-square integer 𝑘≠0. We show that the number of curves 𝐸𝐵:𝑦^2=𝑥^3+𝑘𝐵^2 containing an integral point, where B ranges over positive integers less than N, is bounded by ≪𝑘𝑁(log𝑁)−1/2+𝜖. In particular, this implies that the number of positive integers 𝐵≤𝑁 such that −3𝑘𝐵^2 is the discriminant of an elliptic curve over 𝑄 is o(N). The proof involves a discriminant-lowering procedure on integral binary cubic forms."}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2203.11366"}],"article_processing_charge":"No","publication_identifier":{"eissn":["1432-1807"],"issn":["0025-5831"]},"_id":"19487","date_published":"2023-02-07T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","publication_status":"published","month":"02","type":"journal_article","quality_controlled":"1"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2022-09-01T00:00:00Z","_id":"19490","type":"journal_article","quality_controlled":"1","month":"09","publication_status":"published","oa_version":"Preprint","page":"6675-6700","author":[{"first_name":"Yik Tung","last_name":"Chan","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung"}],"issue":"9","abstract":[{"lang":"eng","text":"Abstract. We study integral points on the quadratic twists ED : y2 = x3 −\r\nD2x of the congruent number curve. We give upper bounds on the number of\r\nintegral points in each coset of 2ED(Q) in ED(Q) and show that their total is\r\n (3.8)rank ED(Q). We further show that the average number of non-torsion\r\nintegral points in this family is bounded above by 2. As an application we also\r\ndeduce from our upper bounds that the system of simultaneous Pell equations\r\naX2 − bY 2 = d, bY 2 − cZ2 = d for pairwise coprime positive integers a, b, c, d,\r\nhas at most  (3.6)ω(abcd) integer solutions."}],"year":"2022","oa":1,"publication_identifier":{"eissn":["1088-6850"],"issn":["0002-9947"]},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2004.03331","open_access":"1"}],"article_processing_charge":"No","scopus_import":"1","article_type":"original","day":"01","language":[{"iso":"eng"}],"publisher":"American Mathematical Society","extern":"1","volume":375,"status":"public","arxiv":1,"OA_type":"green","date_created":"2025-04-05T10:50:56Z","title":"Integral points on the congruent number curve","date_updated":"2025-07-10T11:51:47Z","doi":"10.1090/tran/8732","external_id":{"arxiv":["2004.03331"]},"intvolume":"       375","publication":"Transactions of the American Mathematical Society","citation":{"ieee":"S. Chan, “Integral points on the congruent number curve,” <i>Transactions of the American Mathematical Society</i>, vol. 375, no. 9. American Mathematical Society, pp. 6675–6700, 2022.","chicago":"Chan, Stephanie. “Integral Points on the Congruent Number Curve.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2022. <a href=\"https://doi.org/10.1090/tran/8732\">https://doi.org/10.1090/tran/8732</a>.","ista":"Chan S. 2022. Integral points on the congruent number curve. Transactions of the American Mathematical Society. 375(9), 6675–6700.","short":"S. Chan, Transactions of the American Mathematical Society 375 (2022) 6675–6700.","apa":"Chan, S. (2022). Integral points on the congruent number curve. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/8732\">https://doi.org/10.1090/tran/8732</a>","mla":"Chan, Stephanie. “Integral Points on the Congruent Number Curve.” <i>Transactions of the American Mathematical Society</i>, vol. 375, no. 9, American Mathematical Society, 2022, pp. 6675–700, doi:<a href=\"https://doi.org/10.1090/tran/8732\">10.1090/tran/8732</a>.","ama":"Chan S. Integral points on the congruent number curve. <i>Transactions of the American Mathematical Society</i>. 2022;375(9):6675-6700. doi:<a href=\"https://doi.org/10.1090/tran/8732\">10.1090/tran/8732</a>"},"OA_place":"repository"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2022-05-17T00:00:00Z","_id":"19491","has_accepted_license":"1","quality_controlled":"1","type":"journal_article","month":"05","publication_status":"published","oa_version":"Published Version","abstract":[{"text":"Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let D denote the set of positive squarefree integers having no prime factors congruent to 3 modulo 4 . Stevenhagen [19] conjectured that the density of d in D such that the negative Pell equation x2−dy2=−1 is solvable with x,y∈Z is 58.1% , to the nearest tenth of a percent. By studying the distribution of the 8 -rank of narrow class groups Cl+(d) of Q(√d) , we prove that the infimum of this density is at least 53.8% .","lang":"eng"}],"author":[{"orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","last_name":"Chan","first_name":"Yik Tung"},{"last_name":"Koymans","first_name":"Peter","full_name":"Koymans, Peter"},{"first_name":"Djordjo","last_name":"Milovic","full_name":"Milovic, Djordjo"},{"full_name":"Pagano, Carlo","first_name":"Carlo","last_name":"Pagano"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"oa":1,"year":"2022","publication_identifier":{"issn":["2050-5094"]},"article_processing_charge":"Yes","main_file_link":[{"url":"https://doi.org/10.1017/fms.2022.40","open_access":"1"}],"scopus_import":"1","ddc":["510"],"day":"17","article_number":"e46","language":[{"iso":"eng"}],"article_type":"original","volume":10,"publisher":"Cambridge University Press","extern":"1","date_created":"2025-04-05T10:51:00Z","status":"public","arxiv":1,"OA_type":"gold","date_updated":"2025-07-10T11:51:47Z","title":"The 8-rank of the narrow class group and the negative Pell equation","doi":"10.1017/fms.2022.40","publication":"Forum of Mathematics, Sigma","DOAJ_listed":"1","external_id":{"arxiv":["1908.01752"]},"intvolume":"        10","citation":{"apa":"Chan, S., Koymans, P., Milovic, D., &#38; Pagano, C. (2022). The 8-rank of the narrow class group and the negative Pell equation. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fms.2022.40\">https://doi.org/10.1017/fms.2022.40</a>","ama":"Chan S, Koymans P, Milovic D, Pagano C. The 8-rank of the narrow class group and the negative Pell equation. <i>Forum of Mathematics, Sigma</i>. 2022;10. doi:<a href=\"https://doi.org/10.1017/fms.2022.40\">10.1017/fms.2022.40</a>","mla":"Chan, Stephanie, et al. “The 8-Rank of the Narrow Class Group and the Negative Pell Equation.” <i>Forum of Mathematics, Sigma</i>, vol. 10, e46, Cambridge University Press, 2022, doi:<a href=\"https://doi.org/10.1017/fms.2022.40\">10.1017/fms.2022.40</a>.","short":"S. Chan, P. Koymans, D. Milovic, C. Pagano, Forum of Mathematics, Sigma 10 (2022).","chicago":"Chan, Stephanie, Peter Koymans, Djordjo Milovic, and Carlo Pagano. “The 8-Rank of the Narrow Class Group and the Negative Pell Equation.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2022. <a href=\"https://doi.org/10.1017/fms.2022.40\">https://doi.org/10.1017/fms.2022.40</a>.","ista":"Chan S, Koymans P, Milovic D, Pagano C. 2022. The 8-rank of the narrow class group and the negative Pell equation. Forum of Mathematics, Sigma. 10, e46.","ieee":"S. Chan, P. Koymans, D. Milovic, and C. Pagano, “The 8-rank of the narrow class group and the negative Pell equation,” <i>Forum of Mathematics, Sigma</i>, vol. 10. Cambridge University Press, 2022."},"OA_place":"publisher"},{"title":"A density of ramified primes","date_updated":"2025-07-10T11:51:46Z","doi":"10.1007/s40993-021-00295-5","external_id":{"arxiv":["2005.10188"]},"intvolume":"         8","publication":"Research in Number Theory","citation":{"short":"S. Chan, C. McMeekin, D. Milovic, Research in Number Theory 8 (2021).","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>.","ista":"Chan S, McMeekin C, Milovic D. 2021. A density of ramified primes. Research in Number Theory. 8, 1.","ieee":"S. Chan, C. McMeekin, and D. Milovic, “A density of ramified primes,” <i>Research in Number Theory</i>, vol. 8. Springer Nature, 2021.","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>","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>","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>."},"OA_place":"publisher","scopus_import":"1","article_type":"original","ddc":["510"],"day":"15","article_number":"1","language":[{"iso":"eng"}],"extern":"1","publisher":"Springer Nature","volume":8,"status":"public","OA_type":"hybrid","arxiv":1,"date_created":"2025-04-05T10:50:51Z","author":[{"orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","first_name":"Yik Tung","last_name":"Chan"},{"first_name":"Christine","last_name":"McMeekin","full_name":"McMeekin, Christine"},{"first_name":"Djordjo","last_name":"Milovic","full_name":"Milovic, Djordjo"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"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.","lang":"eng"}],"year":"2021","oa":1,"publication_identifier":{"eissn":["2363-9555"],"issn":["2522-0160"]},"main_file_link":[{"url":"https://doi.org/10.1007/s40993-021-00295-5","open_access":"1"}],"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2021-11-15T00:00:00Z","_id":"19489","has_accepted_license":"1","type":"journal_article","quality_controlled":"1","publication_status":"published","month":"11","oa_version":"Published Version"},{"publication_status":"published","month":"08","oa_version":"Preprint","quality_controlled":"1","type":"journal_article","date_published":"2021-08-17T00:00:00Z","_id":"19492","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1905.09745","open_access":"1"}],"publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"oa":1,"year":"2021","abstract":[{"text":"Kuroda’s formula relates the class number of a multiquadratic number field K to the class numbers of its quadratic subfields ki. A key component in this formula is the unit group index (math formular). We study how Q(K) behaves on average in certain natural families of totally real biquadratic fields K parametrized by prime numbers.","lang":"eng"}],"issue":"2","author":[{"id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","first_name":"Yik Tung","last_name":"Chan","orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung"},{"first_name":"Djordjo","last_name":"Milovic","full_name":"Milovic, Djordjo"}],"page":"1509-1527","date_created":"2025-04-05T10:51:04Z","status":"public","arxiv":1,"OA_type":"green","volume":300,"publisher":"Springer Nature","extern":"1","day":"17","language":[{"iso":"eng"}],"article_type":"original","scopus_import":"1","OA_place":"repository","citation":{"ama":"Chan S, Milovic D. Kuroda’s formula and arithmetic statistics. <i>Mathematische Zeitschrift</i>. 2021;300(2):1509-1527. doi:<a href=\"https://doi.org/10.1007/s00209-021-02823-6\">10.1007/s00209-021-02823-6</a>","mla":"Chan, Stephanie, and Djordjo Milovic. “Kuroda’s Formula and Arithmetic Statistics.” <i>Mathematische Zeitschrift</i>, vol. 300, no. 2, Springer Nature, 2021, pp. 1509–27, doi:<a href=\"https://doi.org/10.1007/s00209-021-02823-6\">10.1007/s00209-021-02823-6</a>.","apa":"Chan, S., &#38; Milovic, D. (2021). Kuroda’s formula and arithmetic statistics. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-021-02823-6\">https://doi.org/10.1007/s00209-021-02823-6</a>","ista":"Chan S, Milovic D. 2021. Kuroda’s formula and arithmetic statistics. Mathematische Zeitschrift. 300(2), 1509–1527.","short":"S. Chan, D. Milovic, Mathematische Zeitschrift 300 (2021) 1509–1527.","chicago":"Chan, Stephanie, and Djordjo Milovic. “Kuroda’s Formula and Arithmetic Statistics.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00209-021-02823-6\">https://doi.org/10.1007/s00209-021-02823-6</a>.","ieee":"S. Chan and D. Milovic, “Kuroda’s formula and arithmetic statistics,” <i>Mathematische Zeitschrift</i>, vol. 300, no. 2. Springer Nature, pp. 1509–1527, 2021."},"publication":"Mathematische Zeitschrift","external_id":{"arxiv":["1905.09745"]},"intvolume":"       300","date_updated":"2025-07-10T11:51:48Z","title":"Kuroda’s formula and arithmetic statistics","doi":"10.1007/s00209-021-02823-6"},{"abstract":[{"text":"In 2016, Balakrishnan, Ho, Kaplan, Spicer, Stein and Weigandt produced a database of elliptic curves over Q ordered by height in which they computed the rank, the size of the 2-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over \r\nQ whose rational torsion subgroup is isomorphic to Z∕2Z×Z∕8Z. Conditional on GRH and BSD, we compute the rank of 92% of the 202,461 curves with parameter height less than 103. We also compute the size of the 2-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.","lang":"eng"}],"author":[{"orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung","first_name":"Yik Tung","last_name":"Chan","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1"},{"last_name":"Hanselman","first_name":"Jeroen","full_name":"Hanselman, Jeroen"},{"full_name":"Li, Wanlin","last_name":"Li","first_name":"Wanlin"}],"page":"173-189","oa":1,"year":"2019","publication_identifier":{"issn":["2329-9061"],"eissn":["2329-907X"]},"article_processing_charge":"No","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1805.10709","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2019-02-13T00:00:00Z","_id":"19493","quality_controlled":"1","type":"journal_article","publication_status":"published","month":"02","oa_version":"Preprint","title":"Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion","date_updated":"2025-07-10T11:51:49Z","doi":"10.2140/obs.2019.2.173","publication":"The Open Book Series","external_id":{"unknown":["1805.10709"]},"intvolume":"         2","citation":{"apa":"Chan, S., Hanselman, J., &#38; Li, W. (2019). Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. <i>The Open Book Series</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/obs.2019.2.173\">https://doi.org/10.2140/obs.2019.2.173</a>","mla":"Chan, Stephanie, et al. “Ranks, 2-Selmer Groups, and Tamagawa Numbers of Elliptic Curves with ℤ∕2ℤ × ℤ∕8ℤ-Torsion.” <i>The Open Book Series</i>, vol. 2, Mathematical Sciences Publishers, 2019, pp. 173–89, doi:<a href=\"https://doi.org/10.2140/obs.2019.2.173\">10.2140/obs.2019.2.173</a>.","ama":"Chan S, Hanselman J, Li W. Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. <i>The Open Book Series</i>. 2019;2:173-189. doi:<a href=\"https://doi.org/10.2140/obs.2019.2.173\">10.2140/obs.2019.2.173</a>","ieee":"S. Chan, J. Hanselman, and W. Li, “Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion,” <i>The Open Book Series</i>, vol. 2. Mathematical Sciences Publishers, pp. 173–189, 2019.","ista":"Chan S, Hanselman J, Li W. 2019. Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. The Open Book Series. 2, 173–189.","chicago":"Chan, Stephanie, Jeroen Hanselman, and Wanlin Li. “Ranks, 2-Selmer Groups, and Tamagawa Numbers of Elliptic Curves with ℤ∕2ℤ × ℤ∕8ℤ-Torsion.” <i>The Open Book Series</i>. Mathematical Sciences Publishers, 2019. <a href=\"https://doi.org/10.2140/obs.2019.2.173\">https://doi.org/10.2140/obs.2019.2.173</a>.","short":"S. Chan, J. Hanselman, W. Li, The Open Book Series 2 (2019) 173–189."},"OA_place":"repository","scopus_import":"1","language":[{"iso":"eng"}],"day":"13","article_type":"original","volume":2,"extern":"1","publisher":"Mathematical Sciences Publishers","date_created":"2025-04-05T10:51:07Z","status":"public","OA_type":"green"},{"citation":{"ista":"Chan S. 2018. Rational right triangles of a given area. The American Mathematical Monthly. 125(8), 689–703.","chicago":"Chan, Stephanie. “Rational Right Triangles of a given Area.” <i>The American Mathematical Monthly</i>. Taylor &#38; Francis, 2018. <a href=\"https://doi.org/10.1080/00029890.2018.1495491\">https://doi.org/10.1080/00029890.2018.1495491</a>.","short":"S. Chan, The American Mathematical Monthly 125 (2018) 689–703.","ieee":"S. Chan, “Rational right triangles of a given area,” <i>The American Mathematical Monthly</i>, vol. 125, no. 8. Taylor &#38; Francis, pp. 689–703, 2018.","apa":"Chan, S. (2018). Rational right triangles of a given area. <i>The American Mathematical Monthly</i>. Taylor &#38; Francis. <a href=\"https://doi.org/10.1080/00029890.2018.1495491\">https://doi.org/10.1080/00029890.2018.1495491</a>","ama":"Chan S. Rational right triangles of a given area. <i>The American Mathematical Monthly</i>. 2018;125(8):689-703. doi:<a href=\"https://doi.org/10.1080/00029890.2018.1495491\">10.1080/00029890.2018.1495491</a>","mla":"Chan, Stephanie. “Rational Right Triangles of a given Area.” <i>The American Mathematical Monthly</i>, vol. 125, no. 8, Taylor &#38; Francis, 2018, pp. 689–703, doi:<a href=\"https://doi.org/10.1080/00029890.2018.1495491\">10.1080/00029890.2018.1495491</a>."},"OA_place":"repository","date_updated":"2025-07-10T11:51:49Z","title":"Rational right triangles of a given area","doi":"10.1080/00029890.2018.1495491","external_id":{"arxiv":["1706.05919"]},"intvolume":"       125","publication":"The American Mathematical Monthly","extern":"1","publisher":"Taylor & Francis","volume":125,"status":"public","OA_type":"green","arxiv":1,"date_created":"2025-04-05T10:51:16Z","scopus_import":"1","article_type":"original","language":[{"iso":"eng"}],"day":"28","publication_identifier":{"issn":["0002-9890"],"eissn":["1930-0972"]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1706.05919"}],"article_processing_charge":"No","page":"689-703","author":[{"orcid":"0000-0001-8467-4106","full_name":"Chan, Yik Tung","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","first_name":"Yik Tung","last_name":"Chan"}],"abstract":[{"text":"Starting from any given rational-sided, right triangle, for example, the (3,4,5)-triangle with area 6, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show further that the set of all such triangles of a given area is finitely generated under our geometric construction. Such areas are known as “congruent numbers” and have a rich history in which all the results in this article have been proved and far more. Yet, as far as we can tell, this seems to be the first exploration using this kind of geometric technique.","lang":"eng"}],"issue":"8","year":"2018","oa":1,"type":"journal_article","quality_controlled":"1","publication_status":"published","month":"09","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2018-09-28T00:00:00Z","_id":"19494"}]