OFFSET

1,1

COMMENTS

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.

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.

REFERENCES

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

LINKS

L. Babinkostova et al., Elliptic Reciprocity, arXiv:1212.1983 [math.NT], 2012.

R. Bröker and P. Stevenhagen, Constructing elliptic curves of prime order, Contemporary Mathematics 463 (2008), 17-28.

R. Bröker and P. Stevenhagen, Constructing elliptic curves of prime order, arXiv:0712.2022 [math.NT], 2007.

J. H. Silverman and K. E. Stange, Amicable pairs and aliquot cycles for elliptic curves, arXiv:0912.1831 [math.NT], 2009.

J. H. Silverman and K. E. Stange, Amicable pairs and aliquot cycles for elliptic curves, Experimental Mathematics , 20:3 (2011), 329-357.

CROSSREFS

KEYWORD

nonn

AUTHOR

Liljana Babinkostova, Jun 18 2016

STATUS

approved