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a(n) is the first term of the n-th proper elliptic 6-cycle.
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%I #16 Feb 05 2023 11:03:08

%S 274723,13415557,27103147,127827253,154689319,162097909,183192157,

%T 196484569,196484569,246836983,246948451,279990229,281840539,

%U 338131501,351159649,392743807,428156821,435821443,459898531

%N a(n) is the first term of the n-th proper elliptic 6-cycle.

%C An elliptic pair over a squarefree integer d is a pair of primes (p,q) such that there exists an elliptic curve E with complex multiplication in the imaginary quadratic field with d<0 which has order q when examined over a prime field of size p.

%C The symbol (p_1, p_2, ..., p_n)_d denotes an elliptic list of length n over d if each of (p_1,p_2), (p_2, p_3), ..., (p_{n-1}; p_n) is an elliptic pair over d. An elliptic cycle is an elliptic list with p_n = p_1. A proper elliptic cycle is one where this is the only equality among terms. The formula for the rest of the terms in a (proper) elliptic 6-cycle is given in Corollary 3.2.in the paper arXiv:1212.1983. Also, some of the results about elliptic pairs and cycles appear in arXiv:1212.1983. The resulting sequence of 6-cycles is based on work done by J. Bahr, Y. Kim, E. Neyman, and G. Taylor.

%D L. C. Washington, Number Theory: Elliptic Curves and Cryptography, Chapman & Hall/CRC, 2nd ed., (2008).

%H L. Babinkostova et al., <a href="http://arxiv.org/abs/1212.1983">Elliptic Reciprocity</a>, arXiv:1212.1983 [math.NT], 2012.

%H R. Bröker and P. Stevenhagen, <a href="http://www.cimpa-icpam.org/archivesecoles/20140205155227/ps2.pdf">Constructing elliptic curves of prime order</a>, Contemporary Mathematics 463 (2008), 17-28.

%H R. Bröker and P. Stevenhagen, <a href="http://arxiv.org/abs/0712.2022">Constructing elliptic curves of prime order</a>, arXiv:0712.2022 [math.NT], 2007.

%H J. H. Silverman and K. E. Stange, <a href="https://arxiv.org/abs/0912.1831">Amicable pairs and aliquot cycles for elliptic curves</a>, arXiv:0912.1831 [math.NT], 2009.

%H J. H. Silverman and K. E. Stange, <a href="http://projecteuclid.org/euclid.em/1317924425">Amicable pairs and aliquot cycles for elliptic curves</a>, Experimental Mathematics , 20:3 (2011), 329-357.

%K nonn

%O 1,1

%A _Liljana Babinkostova_, Jun 18 2016