"소모스-4 수열"의 두 판 사이의 차이

2019년 7월 1일 (월) 02:45 판

점화식으로 정의되는 정수수열
$a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+2}^2$
초기조건이 $a_1=a_2=a_3=a_4=1$ 인 경우
1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786

초기조건이 $a_1=x,a_2=y,a_3=z,a_4=w$ 인 경우\[x,y,z,w,\frac{w y+z^2}{x},\frac{w^2 x+w y z+z^3}{x y},\frac{y(wy+z^2)^2+w x (w^2 x+w y z+z^3)}{x^2 y z}\]
점화식에서 얻어지는 항들이 모두 $\mathbb{Z}[x^{\pm},y^{\pm},z^{\pm},w^{\pm}]$의 원소, 즉 로랑 다항식이며, 이를 로랑현상(Laurent phenomenon) 이라 한다 [FZ2001]

QRT I (Quispel, Roberts, Thompson)\[a_{n+4}a_{n} = \delta^2a_{n+3} a_{n+1} -\delta a_{n+1}^2\]
M. Gaudin "La fonction d'onde de Bethe" (1983)
periodicity => Boltzman weights for hard hexagon model
http://www.google.com/search?hl=en&tbs=tl:1&q=
수학사 연표

The density of primes $p\in \mathbb{Z}$ dividing at least one term of this sequence is $11/21$
Jones, Rafe, and Jeremy Rouse. “Galois Theory of Iterated Endomorphisms.” Proceedings of the London Mathematical Society 100, no. 3 (May 1, 2010): 763–94. doi:10.1112/plms/pdp051.

Van der Kamp, Peter H. ‘Somos-4 and Somos-5 Are Arithmetic Divisibility Sequences’. arXiv:1505.00194 [math], 1 May 2015. http://arxiv.org/abs/1505.00194.
Yura, Fumitaka. “Hankel Determinant Solution for Elliptic Sequence.” arXiv:1411.6972 [math-Ph, Physics:nlin], November 25, 2014. http://arxiv.org/abs/1411.6972.
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
[FZ2001]Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241.

@@ 2번째 줄: / 2번째 줄: @@
 *  점화식으로 정의되는 정수수열
-* <math>a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+1}^2</math>
+* <math>a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+2}^2</math>
 *  초기조건이 <math>a_1=a_2=a_3=a_4=1</math> 인 경우
 *  1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786
 ==로랑현상==