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:<math>\mathrm{H}_8=\frac{3}{10},\,\mathrm{H}_{12}=\frac{567}{130},\,\mathrm{H}_{16}=\frac{43\,659}{170},\,\ldots</math>
Also<ref>{{cite journal |last1=Katz |first1=Nicholas M. |date=1975 |title=The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers |url=https://link.springer.com/article/10.1007/BF02547966 |journal=Mathematische Annalen |volume=216 |issue=1 |pages=1–4|doi=10.1007/BF02547966 }} See eq. (9)</ref>
:<math>\operatorname{denom}\mathrm{H}_{4n}=\prod_{(p-1)|4n}p</math>
where <math>p\in\mathbb{P}</math> such that <math>p\not\equiv 3\,(\text{mod}\,4),</math>
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If <math>a</math> and <math>p</math> are coprime, then there exist numbers <math>p'\in\mathbb{Z}[i]</math> (see<ref>{{cite journal |last1=Eisenstein |first1=G.
|title=Beiträge zur Theorie der elliptischen Functionen |language=German|journal=Journal für die reine und angewandte Mathematik|date=1846 |volume=30| url=https://gdz.sub.uni-goettingen.de/id/PPN243919689_0030?tify=%7B%22pages%22%3A%5B202%5D%2C%22view%22%3A%22scan%22%7D}} Eisenstein uses <math>\varphi=\operatorname{sl}</math> and <math>\omega=2\varpi</math>.</ref> for these numbers) such that<ref>{{cite journal |last1=Ogawa |first1=Takuma |title=Similarities between the trigonometric function and the lemniscate function from arithmetic view point |journal=Tsukuba Journal of Mathematics |date=2005 |volume=29 |issue=1 |doi=10.21099/tkbjm/1496164894 |url=https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.full }}</ref>
:<math>\left(\frac{a}{p}\right)_4=\prod_{p'} \frac{\operatorname{sl}(2\varpi ap'/p)}{\operatorname{sl}(2\varpi p'/p)}.</math>
This theorem is analogous to
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* {{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part IV |publisher=Springer |year=1994 |edition=First |isbn=978-1-4612-6932-8 }}
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* {{cite book |last=Carlson |first=Billie C. |year=2010 |authorlink1=Bille C. Carlson |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor1-link=Frank W. J. Olver |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=19. Elliptic Integrals |chapter-url=https://dlmf.nist.gov/19 |title-link=Digital Library of Mathematical Functions }}
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* {{cite book |last1= Houzel |first1=Christian |author-link1=Christian Houzel |date=1978 |chapter=Fonctions elliptiques et intégrales abéliennes |trans-chapter=Elliptic functions and Abelian integrals |editor-last=Dieudonné |editor-first= Jean |editor-link1=Jean Dieudonné |title= Abrégé d'histoire des mathématiques, 1700–1900. II |publisher=Hermann |pages=1–113 |language=fr}}
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* {{cite journal |last1=Langer |first1=Joel C. |first2=David A. |last2=Singer |year=2011 |title=The lemniscatic chessboard |journal=Forum Geometricorum |volume=11 |pages=183–199 |url=https://forumgeom.fau.edu/FG2011volume11/FG201119index.html }}
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| title = Conformal Projections Based on Elliptic Functions
| ___location = Toronto | publisher = B. V. Gutsell, York University
| series =
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* {{cite journal |last1=Lindqvist |first1=Peter |last2=Peetre |first2=Jaak |authorlink2=Jaak Peetre |title=Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals |journal=The American Mathematical Monthly |date=2001 |volume=108 |issue=5 |pages=403–410 |doi=10.1080/00029890.2001.11919766 |url=https://people.math.osu.edu/lang.162/book/LiPe3.pdf |archive-url=https://web.archive.org/web/20220528124044/https://people.math.osu.edu/lang.162/book/LiPe3.pdf |archive-date=28 May 2022 }}
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*{{cite tech report | last1=McGregor |first1= John L. |date=2005 |title=C-CAM: Geometric Aspects and Dynamical Formulation |institution=[[CSIRO Oceans and Atmosphere|CSIRO Atmospheric Research]] |number=70 |url=https://publications.csiro.au/rpr/pub?pid=procite:eb33ff33-c6d6-4779-8b82-867b2ca6e112 }}
* {{cite book |last1=McKean |first1=Henry |authorlink1=Henry McKean |last2=Moll |first2=Victor |author-link2=Victor Hugo Moll |date=1999 |title=Elliptic Curves: Function Theory, Geometry, Arithmetic |publisher=Cambridge |isbn=9780521582285 |url=https://archive.org/details/ellipticcurvesfu0000mcke |url-access=limited }}
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* {{cite journal |last1=Neuman |first1=Edward |authorlink1=Edward Neuman |year=2007 |title=On Gauss lemniscate functions and lemniscatic mean |journal=Mathematica Pannonica |volume=18 |issue=1 |pages=77–94 |url=http://mathematica-pannonica.ttk.pte.hu/articles/mp18-1/MP18-1(2007)pp077-094.pdf }}
* {{cite journal |last1=Nishimura |first1=Ryo |title=New properties of the lemniscate function and its transformation |journal=Journal of Mathematical Analysis and Applications |date=2015 |volume=427 |issue=1 |pages=460–468 |doi=10.1016/j.jmaa.2015.02.066 |doi-access=free }}
* {{cite journal |last1=Ogawa |first1=Takuma |title=Similarities between the trigonometric function and the lemniscate function from arithmetic view point |journal=Tsukuba Journal of Mathematics |date=2005 |volume=29 |issue=1 |doi=10.21099/tkbjm/1496164894 |url=https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.full }}
* {{cite journal |last1=Peirce |first1=Charles Sanders |authorlink1=Charles Sanders Peirce |date=1879 |title=A Quincuncial Projection of the Sphere |journal=American Journal of Mathematics |volume=2 |issue=4 |pages=394–397 |doi=10.2307/2369491 |jstor=2369491 |url=https://archive.org/details/sim_american-journal-of-mathematics_1879_2/page/n403/mode/2up}}
* {{cite book |last=Popescu-Pampu |first=Patrick |date=2016 |title=What is the Genus? |series=Lecture Notes in Mathematics |volume=2162 |publisher=Springer |doi=10.1007/978-3-319-42312-8 |isbn=978-3-319-42311-1 }}
* {{cite book |last1=Prasolov |first1=Viktor |last2=Solovyev |first2=Yuri |date=1997 |chapter=4. Abel's Theorem on Division of Lemniscate |title=Elliptic functions and elliptic integrals |series=Translations of Mathematical Monographs |volume=170 |publisher=American Mathematical Society. |doi=10.1090/mmono/170 |isbn=978-0-8218-0587-9 }}
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* {{cite book |last1=Reinhardt |first1=William P. |last2=Walker |first2=Peter L. |year=2010a |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=22. Jacobian Elliptic Functions |chapter-url=https://dlmf.nist.gov/22 |title-link=Digital Library of Mathematical Functions }}
* {{cite book |last1=Reinhardt |first1=William P. |last2=Walker |first2=Peter L. |year=2010b |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=23. Weierstrass Elliptic and Modular Functions |chapter-url=https://dlmf.nist.gov/23 |title-link=Digital Library of Mathematical Functions }}
* {{cite arXiv |last1=Robinson |first1=Paul L. |title=The Lemniscatic Functions |date=2019a |
* {{cite arXiv |last1=Robinson |first1=Paul L. |title= The Elliptic Functions in a First-Order System |date=2019b |
* {{cite journal |last1=Rosen |first1=Michael |authorlink1=Michael Rosen (mathematician) |title=Abel's Theorem on the Lemniscate |journal=The American Mathematical Monthly |date=1981 |volume=88 |issue=6 |pages=387–395 |doi=10.2307/2321821 |jstor=2321821 }}
* {{cite book |last1=Roy |first1=Ranjan |title=Elliptic and Modular Functions from Gauss to Dedekind to Hecke |publisher=Cambridge University Press |page=28 |year=2017 |isbn=978-1-107-15938-9}}
* {{cite book |chapter=Some milestones of lemniscatomy |last1=Schappacher |first1=Norbert |author-link1=Norbert Schappacher | date= 1997 |editor1-last=Sertöz |editor1-first=S. |title=Algebraic Geometry |type=Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey |publisher=Marcel Dekker |pages=257–290 | chapter-url=http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf}}
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