"소모스 수열(Somos sequence)"의 두 판 사이의 차이

2019년 7월 1일 (월) 02:52 판

개요

점화식으로 정의되는 수열
소모스 4,5,6,7 은 정수수열이며 소모스 8,9는 정수수열이 아니다
점화식만으로는 정수수열이 되는가가 자명하지 않다
정수수열이 되는가의 문제 (integrality)
합동식을 생각할 때의 주기성 문제 (periodicity modulo n) [Robinson1992]
타원곡선론과 클러스터 대수(cluster algebra) 등의 이론에서 등장

소모스-4 수열

\(a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+2}^2\)

소모스-4 수열

소모스5- 수열

\(a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}\)
1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933

RecurrenceTable[{a[n] a[5 + n] == a[2 + n] a[3 + n] + a[1 + n] a[4 + n], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1}, a, {n, 20}]

소모스-6 수열

\(a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2\)
1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[1]

RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 20}]

http://cis.csuohio.edu/~somos/somos6.html

http://oeis.org/A006722

소모스-8 수열

\(a_{n+8}a_{n} = a_{n+7} a_{n+1} +a_{n+6}a_{n+2}+a_{n+5}a_{n+3}+a_{n+4}^2\)
1,1,1,1,1,1,1,1,4,7,13,25,61,187,775,5827,14815,420514/7,28670773/91,6905822101/2275

RecurrenceTable[{a[n] a[n - 8] == a[n - 1] a[n - 7] + a[n - 2] a[n - 6] + a[n - 3] a[n - 5] + a[n - 4]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1, a[7] == 1, a[8] == 1}, a, {n, 20}]

역사

메모

사전 형태의 자료

관련논문

Fedorov, Yuri N., and Andrew N. W. Hone. “Sigma-Function Solution to the General Somos-6 Recurrence via Hyperelliptic Prym Varieties.” arXiv:1512.00056 [nlin], November 30, 2015. http://arxiv.org/abs/1512.00056.
Gorman, Alexi Block, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse. “The Density of Primes Dividing a Particular Non-Linear Recurrence Sequence.” arXiv:1508.02464 [math], August 10, 2015. http://arxiv.org/abs/1508.02464.
Davis, Bryant, Rebecca Kotsonis, and Jeremy Rouse. “The Density of Primes Dividing a Term in the Somos-5 Sequence.” arXiv:1507.05896 [math], July 21, 2015. http://arxiv.org/abs/1507.05896.
Hone, Andrew N. W. 2010. Analytic solutions and integrability for bilinear recurrences of order six. Applicable Analysis: An International Journal 89, no. 4: 473. doi:10.1080/00036810903329977.
Hone, A. N. W. 2007. Sigma function solution of the initial value problem for Somos 5 sequences doi:0.1090/S0002-9947-07-04215-8
R.W. Gosper, R. Schroeppel, Somos sequence near-addition formulas and modular theta functions, arXiv:math.NT/0703470v1, 15 March 2007.
Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:10.1112/S0024609304004163.
Swart, Christine, and Andrew Hone. 2005. Integrality and the Laurent phenomenon for Somos 4 sequences. math/0508094 (August 4). http://arxiv.org/abs/math/0508094
van der Poorten, Alfred J. 2004. Elliptic curves and continued fractions. math/0403225 (March 14). [2]http://arxiv.org/abs/math/0403225.
[FZ2001]Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241.
[Robinson1992]R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. doi:10.1090/S0002-9939-1992-1140672-5
David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42.

"소모스 수열(Somos sequence)"의 두 판 사이의 차이

2019년 7월 1일 (월) 02:52 판

목차

개요

소모스-4 수열

소모스5- 수열

소모스-6 수열

소모스-8 수열

역사

메모

관련된 항목들

사전 형태의 자료

관련논문

관련도서

둘러보기 메뉴

검색

@@ 1번째 줄: / 1번째 줄: @@
-<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소</h5>
+==개요==
-* [[소모스 수열(Somos sequence)]]
+*  점화식으로 정의되는 수열
+*  소모스 4,5,6,7 은 정수수열이며 소모스 8,9는 정수수열이 아니다
+*  점화식만으로는 정수수열이 되는가가 자명하지 않다
+*  정수수열이 되는가의 문제 (integrality)
+*  합동식을 생각할 때의 주기성 문제 (periodicity modulo n) '''[Robinson1992]'''
+* [[타원곡선]]론과 클러스터 대수(cluster algebra) 등의 이론에서 등장
-<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요</h5>
+==소모스-4 수열==
-* 소모스 4,5,6,7 은 정수수열
+* <math>a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+2}^2</math>
-* 소모스 8,9는 정수수열이 아니다
-*  정수수열이 되는가의 문제 (integrality)<br>
-*  합동식을 생각할 때의 주기성 문제 (periodicity modulo n) '''[Robinson1992]'''<br>
-* [[타원곡선]]론과 클러스터 댓<br>
+* [[소모스-4 수열]]
+==소모스5- 수열==
-<h5 style="margin: 0px; line-height: 2em;">소모스-4 수열</h5>
+* <math>a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}</math>
+*  1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933
-* <math>a_{n+4}a_{n} = a_{n+3} a_{n+2} + a_{n+1}^2</math><br>
+#  RecurrenceTable[{a[n] a[5 + n] == a[2 + n] a[3 + n] + a[1 + n] a[4 + n], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1}, a,   {n, 20}]
-*  초기조건 <math>a_1=a_2=a_3=a_4=1</math> 인 경우<br>
-*  1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786<br>
-#  RecurrenceTable[{a[n] a[n - 4] == a[n - 1] a[n - 3] + a[n - 2]^2,  a[1] == 1, a[2] == 1,   a[3] == 1, a[4] == 1}, a,    {n, 20}]<br>
-* http://oeis.org/A006720<br>
-* http://oeis.org/A006769<br>
-*  초기조건이 <math>a_1=x,a_2=y,a_3=z,a_4=w</math> 인 경우<br><math>x,y,z,w,\frac{w y+z^2}{x},\frac{w^2 x+w y z+z^3}{x y},\frac{y(wy+z^2)^2+w x (w^2 x+w y z+z^3)}{x^2 y z}</math><br>
-*  점화식에서 얻어지는 항들이 모두 <math>\mathbb{Z}[x^{\pm},y^{\pm},z^{\pm},w^{\pm}]</math>의 원소, 즉 로랑 다항식이며, 이를 로랑현상(Laurent phenomenon) 이라 한다<br>
-#  RecurrenceTable[{a[n] a[n - 4] == a[n - 1] a[n - 3] + a[n - 2]^2,<br>   a[1] == x, a[2] == y, a[3] == z, a[4] == w}, a, {n, 10}]<br>
-<h5 style="margin: 0px; line-height: 2em;">소모스5- 수열</h5>
-* <math>a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}</math><br>
-*  1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933<br>
-#  RecurrenceTable[{a[n] a[5 + n] == a[2 + n] a[3 + n] + a[1 + n] a[4 + n], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1}, a,   {n, 20}]<br>
 * [http://www-groups.dcs.st-and.ac.uk/%7Ejohn/Zagier/Solution5.1.html http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.1.html]
-* http://oeis.org/A006721<br>
+* http://oeis.org/A006721
-<h5 style="margin: 0px; line-height: 2em;">소모스-6 수열</h5>
+==소모스-6 수열==
-* <math>a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2</math><br>
+* <math>a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2</math>
-*  1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[http://oeis.org/A006722 ]<br>
+*  1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[http://oeis.org/A006722 ]
-#  RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1,   a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 20}]<br>
+#  RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1,   a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 20}]
-* [http://cis.csuohio.edu/%7Esomos/somos6.html http://cis.csuohio.edu/~somos/somos6.html]<br>
+* [http://cis.csuohio.edu/%7Esomos/somos6.html http://cis.csuohio.edu/~somos/somos6.html]
-* http://oeis.org/A006722<br>
+* http://oeis.org/A006722
-<h5 style="margin: 0px; line-height: 2em;">소모스-8 수열</h5>
+==소모스-8 수열==
-* <math>a_{n+8}a_{n} = a_{n+7} a_{n+1} +a_{n+6}a_{n+2}+a_{n+5}a_{n+3}+a_{n+4}^2</math><br>
+* <math>a_{n+8}a_{n} = a_{n+7} a_{n+1} +a_{n+6}a_{n+2}+a_{n+5}a_{n+3}+a_{n+4}^2</math>
-*  1,1,1,1,1,1,1,1,4,7,13,25,61,187,775,5827,14815,420514/7,28670773/91,6905822101/2275<br>
+*  1,1,1,1,1,1,1,1,4,7,13,25,61,187,775,5827,14815,420514/7,28670773/91,6905822101/2275
-#  RecurrenceTable[{a[n] a[n - 8] ==    a[n - 1] a[n - 7] + a[n - 2] a[n - 6] + a[n - 3] a[n - 5] +<br>     a[n - 4]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1,<br>    a[6] == 1, a[7] == 1, a[8] == 1}, a, {n, 20}]<br>
+#  RecurrenceTable[{a[n] a[n - 8] ==    a[n - 1] a[n - 7] + a[n - 2] a[n - 6] + a[n - 3] a[n - 5] +     a[n - 4]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1,    a[6] == 1, a[7] == 1, a[8] == 1}, a, {n, 20}]
-<h5>재미있는 사실</h5>
+==역사==
-* Math Overflow http://mathoverflow.net/search?q=
-* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
+* http://www.google.com/search?hl=en&tbs=tl:1&q=Somos+sequence
-<h5>역사</h5>
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
 * [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics]
 * [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols]
-* [[수학사연표 (역사)|수학사연표]]
+* [[수학사 연표]]
-<h5>메모</h5>
+==메모==
+* Math Overflow http://mathoverflow.net/search?q=somos
 * http://www.cut-the-knot.org/arithmetic/algebra/SimpleSomosSequence.shtml
 * http://faculty.uml.edu/jpropp/somos.html
 * http://www.math.brown.edu/~jhs/Presentations/ICMSEDSLecture.pdf
-<h5>관련된 항목들</h5>
-<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5>
-* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
-* 발음사전 http://www.forvo.com/search/
-* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
-** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
-* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
-* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
+==관련된 항목들==
+* [[소모스-4 수열]]
-<h5>사전 형태의 자료</h5>
+==사전 형태의 자료==
 * http://ko.wikipedia.org/wiki/
-* [http://en.wikipedia.org/wiki/Somos_sequence ]http://en.wikipedia.org/wiki/Somos_sequence
+* http://en.wikipedia.org/wiki/Somos_sequence
 * http://en.wikipedia.org/wiki/Elliptic_divisibility_sequence
-* http://www.proofwiki.org/wiki/
-* http://www.wolframalpha.com/input/?i=
-* [http://eom.springer.de/default.htm The Online Encyclopaedia of Mathematics]
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-<h5>관련논문</h5>
-* Hone, Andrew N. W. 2010. Analytic solutions and integrability for bilinear recurrences of order six. Applicable Analysis: An International Journal 89, no. 4: 473. doi:[http://dx.doi.org/10.1080/00036810903329977 10.1080/00036810903329977].
+==관련논문==
+* Fedorov, Yuri N., and Andrew N. W. Hone. “Sigma-Function Solution to the General Somos-6 Recurrence via Hyperelliptic Prym Varieties.” arXiv:1512.00056 [nlin], November 30, 2015. http://arxiv.org/abs/1512.00056.
+* Gorman, Alexi Block, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse. “The Density of Primes Dividing a Particular Non-Linear Recurrence Sequence.” arXiv:1508.02464 [math], August 10, 2015. http://arxiv.org/abs/1508.02464.
+* Davis, Bryant, Rebecca Kotsonis, and Jeremy Rouse. “The Density of Primes Dividing a Term in the Somos-5 Sequence.” arXiv:1507.05896 [math], July 21, 2015. http://arxiv.org/abs/1507.05896.
+* Hone, Andrew N. W. 2010. Analytic solutions and integrability for bilinear recurrences of order six. Applicable Analysis: An International Journal 89, no. 4: 473. doi:[http://dx.doi.org/10.1080/00036810903329977 10.1080/00036810903329977].
 * Hone, A. N. W. 2007. Sigma function solution of the initial value problem for Somos 5 sequences doi:[http://dx.doi.org/10.1090/S0002-9947-07-04215-8 0.1090/S0002-9947-07-04215-8]
-* Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:[http://dx.doi.org/10.1112/S0024609304004163 10.1112/S0024609304004163].
+* R.W. Gosper, R. Schroeppel, Somos sequence near-addition formulas and modular theta functions, arXiv:[http://arxiv.org/abs/math/0703470 math.NT/0703470v1], 15 March 2007.
+* Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:[http://dx.doi.org/10.1112/S0024609304004163 10.1112/S0024609304004163].
 * Swart, Christine, and Andrew Hone. 2005. Integrality and the Laurent phenomenon for Somos 4 sequences. math/0508094 (August 4). http://arxiv.org/abs/math/0508094
 * van der Poorten, Alfred J. 2004. Elliptic curves and continued fractions. math/0403225 (March 14). [http://arxiv.org/abs/math/0403225. ]http://arxiv.org/abs/math/0403225.
-* Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241.
+* '''[FZ2001]'''Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241.
 * '''[Robinson1992]'''R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. doi:[http://dx.doi.org/10.1090/S0002-9939-1992-1140672-5 10.1090/S0002-9939-1992-1140672-5]
 * David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42.
-* http://www.jstor.org/action/doBasicSearch?Query=
-* http://www.ams.org/mathscinet
-* http://dx.doi.org/10.1090/S0002-9947-07-04215-8
-<h5>관련도서</h5>
-*  Gale, David. 1998. Tracking The Automatic Ant: And Other Mathematical Explorations. Springer, May 29. <br>  <br>
-*  도서내검색<br>
-** http://books.google.com/books?q=
-** http://book.daum.net/search/contentSearch.do?query=
-*  도서검색<br>
-** http://books.google.com/books?q=
-** http://book.daum.net/search/mainSearch.do?query=
-** http://book.daum.net/search/mainSearch.do?query=
-<h5>관련기사</h5>
-*  네이버 뉴스 검색 (키워드 수정)<br>
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
-<h5>링크</h5>
-*  구글 블로그 검색<br>
+==관련도서==
-** http://blogsearch.google.com/blogsearch?q=
+* Gale, David. 1998. Tracking The Automatic Ant: And Other Mathematical Explorations. Springer, May 29.
-* [http://navercast.naver.com/science/list 네이버 오늘의과학]
+[[분류:수열]]
-* [http://www.ams.org/mathmoments/ Mathematical Moments from the AMS]
-* [http://betterexplained.com/ BetterExplained]
-* [http://www.exampleproblems.com/ exampleproblems.com]